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real group orbits flag apr mikhail ignatyev ivan penkov joseph wolf abstract consider complex real forms main object study arbitrary splitting parabolic assumption subgroups aligned natural way prove intersection flag variety given exhaustion via single also characterize finitely many provide criteria existence open closed case infinitely many keywords homogeneous real group orbit generalized flag ams subject classification introduction study roots linear algebra witt theorem claims given two subspaces vector space endowed nondegenerate bilinear hermitian form spaces isometric within one mikhail ignatyev department mathematics mechanics samara state aerospace university pavlova samara russia ivan penkov jacobs university bremen campus ring bremen germany joseph wolf department mathematics university california berkeley berkeley california usa jawolf mikhail ignatyev ivan penkov joseph wolf obtained via isometry isometric hermitian space statement orbits unitary group complex grassmannian precisely orbits parameterized possible signatures possibly degenerate hermitian form space general theory orbits real form semisimple complex lie group flag variety developed third author theory become standard tool semisimple representation theory complex algebraic geometry automorphic forms automorphic cohomology mention fhw double fibration transforms similar applications representation theory see structure real group orbits cycle spaces applications complex algebraic geometry see example blz fhw finally applications geometric quantization indicated rsw purpose present paper initiate systematic study real group orbits flag precisely generalized flags study classical simple like arose studying stabilization phenomena classical algebraic groups classical lie algebras representations grown separate subfield vast field lie groups lie algebras particular seen classical consist generalized flags rather simply flags general infinite chains subspaces subject two delicate conditions see subsection restrict real forms study arbitrary generalized flags establish several foundational results direction setting assumes certain alignment subgroups first result fact intersected flag variety given exhaustion via yields single means mapping injective using feature able answer following questions finitely many given closed given open answers depend type real form parabolic subgroup instance borel depends choice ordered basis natural representation closed open see results paper first step direction understanding structure real forms classical real group orbits flag splitting parabolic subgroups substantial work lies ahead background section review basic facts real group orbits discuss relevant class lie groups corresponding real forms flag case let complex vector space recall real structure antilinear involution set real form real vector subspace dimr dimc span coincides real form defines unique real structure set fixed point real form complex lie algebra real lie subalgebra real form complex vector space let complex semisimple connected algebraic group real form real closed algebraic subgroup lie algebra real form lie algebra let parabolic subgroup corresponding flag variety group naturally acts third author proved following facts structure see theorems corollary use usual differentiable manifold topology theorem real submanifold number finite iii union open dense unique closed orbit inequality dimr dimc holds theorem relates witt theorem case hermitian form let complex vector space fix nondegenerate hermitian form signature vector space denote group linear operators determinant preserve real form given group naturally acts grassmannian complex subspaces one assign signature restricted form rank positive squares negative ones equals dimension intersection orthogonal complement mikhail ignatyev ivan penkov joseph wolf witt theorem two subspaces belong signatures coincide set min one verify following formula number furthermore subspace open restriction nondegenerate therefore number open orbits equals min unique closed consists subspaces min condition min maximizes nullity form subspaces particular consists totally complex subspaces see details latter case real forms rest paper denotes fixed complex vector space fixed basis fix order via ordered set let denote span dual system definition group group invertible transformations keep fixed finitely many elements difficult verify depends pair clearly operator determinant denote subgroup operators determinant sequel also write instead express basis union nested finite subsets exhausted subspaces hen lim linear operator one assign operator gives embeddings lim follows sider exhaustion fixed set recall inductive limit algebraic varieties manifolds lim always sume form ascending chain closed embeddings endowed topology declaring subset open open follows use terms isotropic totally isotropic synonyms real group orbits flag corresponding topologies morphism lim lim map induced collection morphisms restriction coincides morphism isomorphism exists morphism idy idy morphism induced collection identity maps locally linear algebraic linear algebraic groups inclusions group homomorphisms follows write brevity clearly indsubgroup understand subgroup closed direct limit zariski topology definition real called real form represented increasing union zariski closed subgroups algebraic group real form recall classification real forms due baranov fix real structure invariant denote group invertible determinant operators defined recall linear operator complex vector space real structure defined commutes real structure equivalently maps real form map gives embedding direct limit lim well defined denote real form fix nondegenerate hermitian form suppose restriction nondegenerate denote dimension maximal definite subspace put dimvn let subgroup consisting operators preserving form map induces embedding direct limit lim exists sufficiently large denote real form finally fix quaternionic structure antilinear automorphism assume complex dimension even restriction quaternionic structure furthermore suppose let subgroup consisting linear operators commuting map induces embedding groups denote direct limit lim group also real form next result corollary theorem corollary theorem real forms isomorphism real forms pairwise mikhail ignatyev ivan penkov joseph wolf flag recall basic definitions chain subspaces linearly ordered inclusion set distinct subspaces write subchain immediate successor immediate predecessor also write set pairs immediate successor generalized flag chain subspaces note nonzero vector determines unique pair generalized flag determines see proposition fix linearly ordered set isomorphism ordered sets written write generalized flag called maximal properly contained another generalized flag equivalent condition dim nonzero vectors generalized flag called flag set proper subspaces isomorphic linearly ordered set subset say generalized flag compatible basis exists surjective map every pair form proposition every generalized flag admits compatible basis generalized flag weakly compatible compatible basis set finite two generalized flags weakly compatible exist isomorphism ordered sets subspace dim dim given generalized flag compatible denote set generalized flags endow structure fix exhaustion finite subsets denote hen given denote dim hen dim hen stands cardinality define projective varieties flags hen form subspaces hen dimensions respectively infinite exist infinitely many define embedding real group orbits flag closed embedding algebraic varieties exists bijection inductive limit chain morphisms see proposition section bijection endows structure independent chosen filtration basis explain bijection detail section suppose linear span coincides coincides span dual system assume also inclusion induced exhaustion coincides inclusion defined denote operators diagonal called splitting cartan subgroup fact cartan subgroup terminology dpw define splitting borel parabolic subgroup containing intersection borel parabolic subgroup note splitting parabolic subgroup locally projective indvariety exhausted projective varieties one easily check group naturally acts given generalized flag compatible denote stabilizer proof following theorem see proposition theorem theorem let generalized flag compatible group parabolic subgroup containing map bijection generalized flags compatible splitting parabolic subgroups fact map induces isomorphism iii maximal splitting borel subgroup example first example generalized flags provided flag proper nonzero subspace compatible assume subset case called denoted dim finite flag dim hence depends denote similarly codim finite depends isomorphic isomorphism dim induced map finally infinite dimensional infinite codimensional depends isomorphic denoted see section details clearly case one mikhail ignatyev ivan penkov joseph wolf second example generalized flag clearly flag flag dim dim fei fei large enough flag maximal iii put odd generalized flag clearly flag maximal imply fei large enough note also example let fei odd remark examples stabilizer precisely maximal splitting parabolic flag equals borel realization section establish basic property orbits real form precisely prove intersection single orbit consequently real start describing explicitly bijection lim mentioned section let generalized flag compatible basis corresponding generalized flags recall consider inductive limit flag varieties embeddings defined previous subsection put construction reformulated follows dimensions spaces flag form sequence integers dsm dim let flag variety type since either unique case case words minimal nonnegative integer dim dim real group orbits flag define embedding given flag gsm set choose positive integer compatible bases containing vmg contains subspace makes generalized flags addition assume fact set compatible let arbitrary sequence integer numbers denote vmn dmn vmn according bijection lim subsection form lim slight abuse notation sequel denote canonical embedding letter let real form see theorem group naturally acts map equivariant put also real form rest paper fix specific assumptions different real forms describe assumptions case case let recall restriction fixed nondegenerate hermitian form nondegenerate assume orthogonal respect next let assume odd hen real form finally assume even additional assumptions align real form flag variety main result section follows theorem nonempty intersection intersection single roof proof goes case case ase proof completely similar pick two flags asmn bsmn belong given mikhail ignatyev ivan penkov joseph wolf put esm bes bei exists satisfying prove result must construct isometry satisfying course one scale obtain isometry determinant huang extension witt theorem theorem isometry exists isometric smn dim dim smn denotes complement within subspace pick smn since orthogonal bei first condition satisfied establishes isometry remains prove denote since orthogonal given subspaces one hence bek subspaces similar equality holds bei result follows ase first prove flags images map exists exists operator satisfying consider first case denote assume maps afto since vector belongs let defined clearly put one real group orbits flag bei bei hence cases operator maps may assume without loss generality operator projection onto along invertible defined maps required suppose case condition contradicts equality arguing see one construct case instead therefore may assume let linearly independent choose define follows put one easily check satisfies required conditions ready prove result namely let satisfy belongs hence exists maps continuing process see exists operator since one scale obtain required operator ase let two flags let satisfy goal conb struct repeated application procedure imply result simplicity denote recall note commutes first suppose belong vector admits unique representation form set det let operator defined mikhail ignatyev ivan penkov joseph wolf easy see commutes det dett furthermore one check commutes maps projection onto along since det one scale obtain operator required second suppose belong one argue first case instead may assume without loss generality note linearly dependent case denote subspace spanned basic vectors linearly independent choose define following rule one check det det commutes maps thus satisfies required conditions following result immediate corollary theorem corollary let single real roof suppose exists images morphism belong applying theorem subsequently see belong definition lim next implies real theorem dimr dimc since dimc conclude infinite dimensional case finitely many give criterion finite number observe case degeneracy order coincides large enough recall degeneracy order orbits partial order generalized flag finite consists finitely many possibly infinitedimensional subspaces say generalized flag finite type consists finitely many subspaces either finite dimension finite codimension finite type generalized flag clearly flag finite type finite type equivalently finite type proposition number orbits finite finite type real group orbits flag number finite finite number equals roof ase first consider case subspace clearly finite type dim codim note construction simply following given let span fixed subspace dim embedding form recall codim map induces isomorphism dim denote isomorphism operator one assign linear operator acting defines isomorphism becomes isomorphism hence finite type consider case dim dim consists subspaces pick exists witt theorem shows belong signatures forms coincide since conclude belong signatures coincide thus number finite hand dim codim lim lim dimvn case number possible signatures restriction dimensional subspace tends infinity hence number tends infinity theorem number infinite consider general case let finite type finite type subflags consisting finitedimensional subspaces respectively note compatible basis hence exists codim codim set dim codim denote signature subspace according theorem check number mikhail ignatyev ivan penkov joseph wolf finite enough prove following sets finite dim codim dim dim dim dim codim codim pab dim dim codim finiteness obvious particular implies number finite applying map described see number finite consequently sets finite finally since nondegenerate see codim dim codim hence pab finite thus finite type number finite hand suppose finite type space dim codim done map subspace corresponding epimorphism number infinite finite dimension finite codimension exist subspaces arbitrarily large dimension arbitrarily large codimension former case statement follows fact number possible signatures spaces tends infinity latter case statement gets reduced former one via map ase first suppose finite given denote dima let subspace clearly hence hand subspaces dim thus theorem shows number less equal hence theorem number finite suppose infinite case given exists length flag less positive index dimension maximal positive definite subspace equals codim easy check number less consequently theorem number less proof complete real group orbits flag ase evident ase first let subspace compatible dim consists subspaces claim number equals indeed pick recall dimvn clearly belong dim dim since dim arbitrary integer number hence number least hand suppose satisfied let complex subspaces respectively clearly furthermore easy see simplicity set define similarly note subspaces defined lemma open corresponding grassmannian furthermore two open grassmannian union single hence exists operator defined maps respectively since defined exist complements thus one extend operator finally since dimvn odd scale obtain operator maps required contrary assume dim shown given two spaces belong dim dim number grassmannian equals number infinite theorem consider general case claim given type exists number number flag variety less equal upper bound depends dimension prove denote subgroup preserving bilinear form matsuki duality exists correspondence set set hence claim follows mikhail ignatyev ivan penkov joseph wolf diately theorem holds nondegenerate symmetric bilinear forms finally suppose finite type let note form nondegenerate hence complement subspace dimension codim arguing applying remark matsuki duality conclude exists number number less equal every follows theorem total number also less equal finally finite type case one use projections onto show number infinite proof complete ase denote antisymmetric bilinear form defined let subgroup preserving form maximal compact subgroup see fhw duality given exists bijection set set flag variety since isomorphic argue complete proof example let follows formula number grassmannian see subsection proved proposition proof similar case corollary theorem describe degeneracy order set arbitrary finite type definition define partial order set similar way corollary suppose number finite exists isomorphic partially ordered set roof given exists nonempty since finitely many exists real group orbits flag nonempty orbit theorem given exists unique hence map well defined clear map bijective remains note definition topology contained closure contained closure thus fact isomorphism partially ordered sets open closed orbits section provide necessary sufficient conditions given open closed also prove open closed orbit number orbits finite first consider case open orbits pick recall flag open nondegenerate respect dim minimal equals max dim dima dimvn dim minimal sense note two generalized flags canonical identification linearly ordered sets space call image identification space corresponding fix antilinear operator point general position respect properly contain spaces corresponding respectively similar definition given flags note open general position respect respectively case mind give following definition generalized flag nondegenerate nondegenerate respect general position respect respectively mikhail ignatyev ivan penkov joseph wolf remark generalized flag nondegenerate respect thought general position respect therefore conditions definition clearly analogous proposition open nondegenerate roof definition topology open open first suppose prove claim case suffices show nondegenerate respect nondegenerate straightforward indeed degenerate exists let degenerate hand orthogonal orthogonal orthogonal result follows second consider case suppose open open satisfying assume nondegenerate exist subspaces corresponding respectively let since ben ben means ben properly contained hence general position respect contradicts condition open assume open means exist gen properly conen ben ben respective subspaces gen corresponding tains since space properly contains respective images embedding repeating procedure see nondegenerate result follows case considered similarly say two generalized flags type automorphism transforming one clearly two generalized flags always type hand clearly true two generalized flags type basis turns requirement existence open orbit form imposes restriction type flag precisely corollary always orbit exist basis generalized flag type open roof let positive integer positive index equals let flag consisting nondegenerate real group orbits flag subspaces open denote linear operator clearly belongs nondegenerate therefore open consider case let basis fix bijection defines automorphism denote bijection generalized flag consisting images subspaces isomore type space nondegenerate phism spanned subset thus open open remark course general equal given situation different open orbit either dim codim type flag exame form ple open orbit long basis satisfies indeed suppose pointed example exists fbn fen een pick compate integer max xen ible respect bases containing flag general posicontains subspace fbn defined thus tion respect open consequently open generalized flag type let generalized flag example iii recall odd claim also open orbit indeed assume compatible basis finite means exists subspace pick max dim general position respect finally let case clearly may may open orbit similar argument shows example open orbit surprisingly example iii may open orbit consider first case easy check dimvn general position respect orbit open type subspace hand spanned vectors open orbit generalized flag contains subspace mikhail ignatyev ivan penkov joseph wolf turn attention closed orbits conditions orbit closed based idea real forms case open orbits details differ suppose call generalized flag pseudoe isotropic properly contained subspaces corresponding respectively similar definition given flags isotropic generalized flag defined always converse hold particular case generalized flag form kernel form maximal flags form next suppose generalized flag real condition turns equivalent following condition properly contained subspaces corresponding respectively finally suppose call generalized flag pseudoe quaternionic properly contained subspaces corresponding respectively quaternionic clearly pseudoquaternionic converse hold generalized flag form codim proposition closed real roof first consider case unique closed see theorem real forms conditions proposition applied flags easily checked closed conditions points therefore flag closed flag satisfies conditions proposition finite level let suppose closed closed satisfying assume exist subspaces corresponding respectively let ben means properly contained hence contradicts condition closed assume closed means exist gen properly conn tained subspaces corresponding real group orbits flag respectively since orthogonal properly respective imcontained ben embedding repeating procedure ages see result follows let given denote note given defined closed subspace defined hence closed closed vice versa proof similar case based following facts subspace subspace corollary always closed orbit roof generalized flag closed basic vectors choice clearly may closed orbit hand assume example exists orthogonal basis fbn fbn nondegenerate obviously exists thus contain degenerate closed orbit let obviously may may generalized flag closed orbit type generalized flag example satisfies reb quired conditions closed orbit indeed assume contains certain form nondegenerate hence isotropic exists isotropic subspace dimension dim fbn conb taining fbn easy see exists subspace corresponding fbn thus suppose example iii may may closed orbit example assume basis contains nonisotropic subspace arguing previb hand suppose ous paragraph see orthogonal basis case one easily check closed finally let three cases iii example type may generalized flag may closed orbit consider instance case flag closed hand spanned subspace closed orbit mikhail ignatyev ivan penkov joseph wolf combining results existence open closed orbits obtain following corollary corollary given real form generalized flags open closed finitely many roof finitely many existence open orbit obvious existence closed orbit follows immediately corollary assume open closed let fix nondegenerate generalized flag lying open orbit suppose exists subspace satisfying dim codim since closed exists generalized flag let subspace corresponding since commensurable exists subspaces span certain infinite subset particular subspace restriction nondegenerate orthogonal hence implies clearly case exists contained one easily construct generalized flag corresponding subspace contradiction thus contradicts condition codim conclude subspace finite dimension finite codimension assume finite least one generalized flags infinite suppose infinite case infinite considered using map let compatible bases containing let subspace belong arguing previous paragraph contradiction one show let suppose general position respect real pick compatible bases containing suppose moment exists subspace subspace nonzero subspace spanned subset similarly corresponding subspace form suppose also exist let spanned easy check exists subspace corresponding form thus properly contradiction remains note finite type contains subspace always exists necessary applying map finally let suppose general position ree pick spect compatible bases containing suppose moment exists subspace subspace nonzero subspace spanned subset real group orbits flag form subspace corresponding subspace suppose also dim codim exist subspace subspace corresponding form either properly contains properly contained thus eie ther general position respect contradiction remains note finite type subspace always exists possibly applying map acknowledgements first author supported part russian foundation basic research grants dynasty foundation ministry science education russian federation project part work done oberwolfach research institute mathematics program oberwolfach leibniz fellows jacobs university bremen first author thanks institutions hospitality second third authors thank professor dobrev invitation speak international workshop lie theory applications physics varna june second author acknowledges continued partial support dfg priority program spp grant third author acknowledges partial support dickson emeriti professorship university california simons foundation collaboration grant mathematicians references baranov finitary simple lie algebras algebra blz barchini leslie zierau domains holomorphy representations manuscripta math barlet koziarz fonctions holomorphes sur espace des cycles intersection math research letters bremigan lorch matsuki duality flag manifolds manuscripta math dimitrov penkov generalized flags homogeneous spaces classical imrn dimitrov penkov locally semisimple maximal subalgebras finitary lie algebras algebra dpw dimitrov penkov wolf theory direct limits algebraic groups amer math fels huckleberry characterization cycle domains via kobayashi hyperbolicity bull soc math france fhw fels huckleberry wolf cycle spaces flag domains complex geometric viewpoint progr math boston fresse penkov schubert decomposition generalized flags asian appear see also arxiv huang extensions witt theorem linear multilinear algebra huckleberry simon cycle spaces flag domains sln appendix barlet reine angew math huckleberry wolf cycle spaces real forms sln complex geometry springer verlag berlin huckleberry wolf injectivity double fibration transform cycle spaces flag domains lie theory mikhail ignatyev ivan penkov joseph wolf stanton holomorphic extensions representations automorphic functions ann math stanton holomorphic extensions representations geometry harmonic analysis geometric functional analysis wolf geometry borel siebenthal discrete series lie theory penkov tikhomirov linear pure applied math quarterly rsw rawnsley schmid wolf singular unitary representations indefinite harmonic theory functional analysis schmid wolf geometric quantization derived functor modules semisimple lie groups functional analysis wells wolf series automorphic cohomology flag domains ann math wolf action real semisimple lie group complex flag manifold orbit structure holomorphic arc components bull amer math soc wolf fine structure hermitian symmetric spaces boothby weiss eds symmetric spaces marcel dekker new york wolf action real semisimple group complex flag manifold unitary representations partially holomorphic cohomology spaces memoirs american mathematical society wolf orbit method nondegenerate series hiroshima math wolf completeness series automorphic cohomology ann math wolf admissible representations geometry flag manifolds contemp math wolf zierau holomorphic double fibration transforms proceedings symposia pure mathematics
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dec lipschitz solvable lie groups david bruce cohen february abstract space said lipschitz every lipschitz loop bounds disk lipschitz space admits quadratic isoperimetric inequality unknown whether converse true cornulier tessera showed certain solvable lie groups quadratic isoperimetric inequalities extend result show groups lipschitz introduction lipschitz consider complete simply connected riemannian manifold homogeneous sense lie group acts transitively isometries well known manifold every loop admits lipschitz filling see proposition much stronger result see corollary classical way study large scale geometry asking hard fill loops definition let loop filling span denoted defined inftlippf say lipschitz exists constant lipschitz loops instance euclidean space lipschitz one may cone loops given loop may take obtain filling similar argument shows cat manifolds lipschitz solvable groups paper interested case solvable real lie group equipped left invariant metric motivate note lipschitz proposition every lie group quasi isometric solvable lie group assume form following conditions hold abelian lie group closed subgroup group real upper triangular matrices diagonal entries equal contractible see examples groups lipschitz groups sol type fix real numbers consider group matrices form note decomposes take diagonal matrices matrices diagonal entries equal known theorem exists loops minimal area filling order hence lipschitz groups form called groups sol type generally cornulier tessera shown group surjects onto group sol type loops exponentially large area hence lipschitz theorem surjects onto group sol type say sol obstruction tame groups hand conjugation action contracts lipschitz proposition contraction means compact subset compact subset positive power conjugates exist said tame theorem cornulier tessera clear space lipschitz admits quadratic isoperimetric loop length must bound disk area cornulier tessera given large class solvable lie groups admitting quadratic isoperimetric inequalities shall extend result show groups lipschitz given action abelian group vector space let subspace consisting vectors log theorem theorem states satisfies quadratic isoperimetric inequality following conditions hold said standard solvable condition holds surject onto group sol type lie algebra denotes second lie algebra definition killing module killpuq quotient symmetric square spanned elements form main theorem primary objective paper establish following theorem proved theorem improving cornulier tessera result show lipschitz theorem let group form contractible real lie groups abelian real unipotent group closed group strictly upper triangular real matrices trivial surject onto group sol type lipschitz quadratic isoperimetric inequality versus lipschitz noted lipschitz quadratic isoperimetric inequality known whether converse known examples quadratic isoperimetric inequality lipschitz recently lytchak wenger young shown results existence holder fillings spaces admitting quadratic isoperimetric inequalities organization paper organized follows recalls known results lipschitz filling homogeneous manifolds develops combinatorial language describing fillings lie groups specializes solvable lie groups reviews theory tame groups abels multiamalgam finally prove main theorem acknowledgments wish thank robert young yves cornulier productive conversations work supported nsf award preliminaries filling section complete simply connected riemannian manifold admitting transitive lie group action isometries collect several facts filling loops everything section trivial well known seems easier write proofs find literature key facts prove follows proposition shows lipschitz small scale constant corollary shows loops lipschitz fillings proposition proves existence filling span function function cpx cpx lipschitz loops proposition shows lipschitz setting templates let unit disk equipped euclidean metric purely matter convenience think unit circle subsets need convenient cellular decompositions figure right decomposition unit disk triangles uniformly bilipschitz proposition right decomposition central disk together sectors uniformly bilipschitz proposition definition let euclidean disk radius proposition exists constant triangulation triangles cbilipschitz see left side figure decomposition type proof proposition left reader using templates proposition help fill loops following way suppose class loops consist loops small lipschitz constant given loop may wish find filling often use following abstract strategy construct sufficiently small take decomposition proposition decomposition let map universal constant guaranteed proposition take extend decomposition way lippf restriction boundary represents element want every map given satisfies observe exists filling extend lipschitz constant taking pzq pzq nontrivial part type argument comes try extend desired properties typically suppress details proposition exists constant unit disk may cellularly decomposed inner disk radius surrounded annular region divided call sectors following properties sector bounded two radial line segments arc arc boundary inner disk sectors number sectors see right side figure decomposition type proof proposition left reader proposition typically used convert filling loop filling nearby loop taking restricted inner disk slightly rescaled filling basic results filling homogeneous manifolds proposition let simply connected complete homogeneous riemannian manifold exist constants cpxq dpxq following hold dpx dpxq unique geodesic connecting suppose dpxq cpxq proof clear uniform upper bound sectional curvatures since homogeneous therefore cat space theorem implies proposition constant depending points separated less connected unique geodesic taking dpxq ensures unique geodesics existence normal coordinates riemannian manifolds proposition exists bilipschitz map neighborhood neighborhood rdimpxq take small enough pxq pxqq convex set let without loss generality may fill coning define filling letting let filling lippf taking dpxq cpxq result follows corollary lipschitz maps proof let continuous filling lipschitz must find lipschitz filling let triangulation given proposition let map guaranteed proposition let largest value adjacent vertices proposition lets produce lipschitz filling follows set pxq vertex small enough dpxq constant given proposition dpxq constant given proposition set constant speed geodesic edge map given proposition admits lipschitz extension hand every natural number may find points dpxn fixed compact know pxn must subconverge point diagonal subconverge contradicts continuity proposition let simply connected complete homogeneous riemannian manifold exists cpx cpx gromov refers cpx filling span function proof fix constants presumed depend uncontrollably let denote loops fixed basepoint equip uniform metric compact certainly span continuous show bounded compact every may covered finite number balls hence suffices find constant let dpxq proposition assuming without loss generality dpxq let dpxq let decompose lipschitz filling let dpxq inner disk sectors proposition produce filling lippf inner disk annular region desired define equal let radius inner disk define rescaled copy inner implies lippf inner disk images endpoints radial segment separating two sectors dpx dpxq dpxq define minimal speed geodesic radial segments separating two sectors boundary sector considering fixed since sectors uniformly bilipschitz admits extension sector proposition see annular region lippf inner disk implying desired bound taking sufficiently large proposition let simply connected complete homogeneous riemannian manifolds suppose lipschitz seems likely bilipschitz proof let quasi isometry quasi inverse let loop without loss generality let obtain filling proceed follows using proposition subdivide triangles bilipschitz adjacent vertices boundary mapped within let loop agrees vertices triangulation assumption lipschitz admits filling wish convert filling let map agrees vertices map exists may fill edges constant speed geodesics fill triangles proposition maps adjacent vertices within gives filling note distance uniform metric agrees vertices take new subdivision using proposition subdivide inner disk surrounded oplq sectors uniformly bilipschitz build filling follows inner disk let given rescaled copy radial segments take constant speed geodesic fill sectors using proposition lipschitz moves specialize case equal simply connected lie group equipped riemannian metric main goal section reduce questions filling loops questions manipulating words compact generating set notation simply connected lie group admits compact generating set set consists words together empty word length word denoted given denotes word represents identity element said relation represent element write word norm respect denoted define assumptions time take compact generating set assume without loss generality unless otherwise indicated suffices fill unit square convenient consider loops maps rather fillings maps unit square rather definition given filling unit square map following properties agrees bottom edge unit square meaning constant set three edges unit square meaning define infimum lippf ranges lipschitz fillings unit square similarly given homotopy map restrictions constant see makes difference whether consider loops maps key tool following lemma lemma exists constant given loops reparameterization given filling maxtlippf proof equivalent statement proved lemma corollary universal constant following property suppose lipschitz loop lipschitz path proof let equal bottom edge three edges fix bilipschitz map observe reparameterization lipschitz filling lemma oplippf filling oplippf filling see reverse inequality proved similarly filling relations suffices fill words let compact generating set choose lipschitz curve connecting following properties hold uniform bound ranges constant ptq given word let denote concatenation paths reparameterized paths used ptq proposition suppose exists constant every exists filling unit square lipschitz proof note compact hence baire category theorem power must contain open neighborhood contains open neighborhood identity turn contains around identity take natural number contains around identity larger given loop produce filling follows let propositions suffices consider least cellularly decompose together collection unit square rectangle squares runs define constant vertical edges note element lies may represented horizontal edge connecting let copy translated take ptq letting observe agrees bottom rectangle may extended rectangle lippf first bound comes hypothesis finally admits extension square proposition definition suppose collection pairs words say exists constant depending following property given map equal bottom edge top edge constant sides admits filling give basic rules manipulating fillings often set inferred context lemma following rules hold simply connected lie group suppose lippwq oplippvqq pair fix suppose given sets pvn given pairs exists let bounded neighborhood identity collection relations every prefix represents element lipschitz proof first item follows immediately lemma prove second item subdivide unit square rectangles form suppose given pvn hypothesis exists homotopy lippfi oplippv define map putting rescaled copy take thus lippf opmaxtn lippfi nuq oplippvqq fixed since restricted top unit square reparameterization restricted bottom unit square reparameterization done proof divide unit square two item similar third rectangles put rescaled copy homotopy bottom rectangle rescaled copy homotopy top rectangle get desired homotopy prove fourth item let let defined mintt suuq reader may check filling prove fifth item let contains given let write find filling together follows subdivide rectangle define constant squares form choose take vertical edge ptq may extend square lipschitz constant proposition sixth item consequence proposition normal form triangles discuss normal forms using technique gromov shall see lipschitz lipschitz definition normal form compactly generated group equipped compact generating set map normal form word form lemma let normal form lipschitz proof proved proposition proof sketched figure tame subgroups multiamalgam assumptions specialize case contractible lie groups abelian closed group strictly upper triangular real matrices standard solvable groups observe acts abelianization fix norm vector space vector log say standard solvable section interested structure geometry standard solvable groups describe standard tame subgroups standard solvable group lemma show find lipschitz fillings words already represent identity free product standard tame subgroups theorem give conditions may presented free product standard tame subgroups modulo certain easily understood amalgamation relations figure figure indicates fill arbitrary relation given one knows fill top edge vertical edges taken constant horizontal edge understood appropriate translate label rectangle represents except bottom row bottom edge taken row squares along bottom may filled proposition weights let standard solvable let lie algebra identified usual tangent space fix norm denote conjugation action adpaq expptxqa observe hompa vector space adpaqx definition see homomorphism define space consist together lim log adpaqn define set weights consist hompa dim conic subset mean intersection open convex cone hompa contain denote set conic subsets let denote closed connected subgroup whose lie algebra let groups referred standard tame subgroups see remarks exercise reader may wish compute weights weight spaces group sol type finite define puq recall definition killpuq able state theorem main theorem maps definition let modules induced taking define puq define killpuq quotient symmetric square subspace spanned elements form puq observe natural aaction descends puq identity adpaqrx radpaqx adpaqy subspaces preserved action similarly killpuq also recall define consist together vectors log thus define subspaces tame subgroups definition given vacuum subset compact every compact adpaqn say tame exists vacuum subset tame tame proposition wish show tame lipschitz starting point following proposition suppose tame compact generating set adpaqs proof hypothesis vacuum subset let compact generating set proof proposition see power contains open ball around identity hence set contains vacuum set exists adpbql taking generating set contains standing assumption generating sets contain identity adpaqs adpbql desired proposition tame lipschitz remark probably tame group cat choice metric know prove give combinatorial proof using lemma proof fix compact generating set proposition let generating set let generating set note adpaqpsq observe suj psu function satisfies natural numbers words given exist follows letting denote projection may define normal form word given minimal length word representing word form oplog check normal note opexp oplogpopexp qqq exponentially distorted proposition suffice show form satisfying show thus suffices establish following two facts ubq lemma crucial showing first particular use lemma provide homotopies words stay inside bounded neighborhood identity conjugation show ubq note may written adpaqs adpaqc let let given word adpaq homotope follows liberally using lemma bak ubq filling show write aki let let adpaki qsi homotope follows pak qpak qpak pak qpak qpak filling freely trivial words return case standard solvable group necessarily tame recall collection conic subsets finite let tame group let sgc compact generating set group let let union sgc natural map lemma represents identity image show admits filling need following auxiliary results reader probably skip directly proof lemma understand point propositions representing identity proposition shows word may reduced identity repeated deletion subwords represents identity gcj proposition describes appropriately lipschitz rectangular homotopy words obtained deletions proposition describes part homotopy given proposition exists proposition given word natural number words following properties gcj proof note element lies represents identity theory free products nonempty subword comprised elements sgc represents identity thus may write desired applying argument recursively obtain desired proposition given natural number exists opk lipschitz map following properties along bottom given meaning ptq along top given meaning ptq constant sides depend proof let given follows ptq observe opk may take desired filling proposition given relation natural number let exists map following properties along bottom given meaning ptq along top given meaning ptq constant sides depend proof see right hand side figure subdivide rectangle define ptq first extend top rectangle ptq ptq set constant define ptq extend simply apply proposition thus given extension top rectangle extend bottom rectangle define ptq finally apply proposition extend since equivalent filling lemma recalling notation introduced start relation take sequence proof see figure let words proposition define filling follows let subdivide rectangles set ptq noting proposition shows may extended rectangle lipschitz constant distortion see standard solvable elements may expressed much efficiently generators generators proposition suppose standard solvable compact generating set compact generating set exists proof follows effort proposition proposition prop figure figure left depicts strategy filling freely trivial word proposition figure right depicts proof proposition allows fill rectangle left hand figure multiamalgam subsection define multiamalgam standard solvable group first introduced abels quote key theorem cornulier tessera states certain conditions means put together standard tame subgroups nice way eventually let build fillings fillings standard tame subgroups order state theorem proper generality must briefly discuss theory unipotent groups unipotent groups commutative real unipotent group closed group upper triangular real matrices diagonal entries equal theory algebraic groups allows define group ppq particular glpn consists upper triangular matrices diagonal entries equal ppq consists upper triangular matrices diagonal entries equal matrices certainly invertible determinant equal suppose consists functions obvious bijection ppq ppq may speak definition see let real standard solvable multiamalgam standard tame subgroups defined xxrg tic puq denotes inclusion direct product similarly multiamalgam defined xxru tic puq denotes inclusion direct product commutative define ppq similarly ppq understood ppq course recall admits sol obstruction surjects onto group sol type cornulier tessera give conditions isomorphic theorem let standard solvable real lie group admit sol obstruction ppq ppq commutative proof follows corollary hypotheses corollary satisfied proposition proof main theorem rest paper devoted proof following theorem theorem let contractible real lie groups abelian real unipotent group closed group strictly upper triangular real matrices standard solvable surject onto group sol type trivial lipschitz proof lemma show exists generating set normal form certain properties lemma show properties lemma suffice prove theorem defining standing assumptions throughout rest paper assume satisfies hypotheses theorem standard solvable group surject onto group sol type notation let rps ppq commutative algebra let rps denote inclusion write rrs let compact generating set let compact generating set let cpc theorem compact generating set let compact generating set let generating set equal union compact generating sets ranges let shu union generating set given let psa minimal length word representing let projection let set theoretic map defined pgq pgq lemma standing assumptions exists finite sequence conic subsets normal form following properties pgqq minimal length word representing form uci proof follows proposition give different proof order introduce trick used later trick set define commutative ralgebra collection functions note element ppy may identified function alternatively one may think element ppy family functions indexed matrix coefficients uniformly bounded polynomial choose least continuum cardinality let ppy every certainly possible instance one might take set claiming ppy used fact matrix coefficients polynomial since theorem ppy generated union ppy may write element uci ppy observe exists let element definition exist let uci ppy take note oplog ysci oplog oplog thus defined elements desired properties representing define taking shortest word extend setting pgqq must show normal pgqqq suffices show pgqqq minimal length word representing note represents pgq constant depending log thus since oplog pgqq desired filling wish show may fill triangles normal form produced lemma proposition allow homotope relations form word sci efficiently representing element uci fixed sequence conic subsets order fill relations recall theorem standing assumptions ppq ppq commutative consequently kernel ppq normally generated elements form puq ppq order fill corollary need factor free product product bounded number elements form puqg product bounded number elements living factors controlled polynomial lemma see lemma suppose standing assumptions satisfied given sequence conical subsets exist natural numbers sequence uci equality form pgn satisfying gjk ucjk cjk form puj puj conical subsets proof special case lemma reprise details use cornulier tessera trick used prove lemma take proof lemma recall ppy may thought function let set least continuum cardinality exists ppy uck ppy following strong surjectivity property ppy uck ppy exists theorem know equality form rpy ppy form conical subsets ppy ppy since ppy generate rpy must lives ucjk cjk note definition given uck let sci choose let gjk puj puj follows pgn satisfy desired conditions corollary suppose standing assumptions satisfied given sequence conical subsets sequence uci proof lemma independent pgn gjk ucjk form puj puj conical subsets let let equal minimal length word representing minimal length word representing implies represents first show note conic subset psa represents thus proposition next observe log oplog proposition similarly pgn pgn lemma lemma pgn pgn noted need one proposition fill given let denote proposition fix sequence conical subsets uci proof proposition conclude proof main theorem showing may fill lemma standing assumptions proof recall form sci fixed sequence conical subsets let let pgi let expanding applying proposition repeatedly slide right see resulting word admits lipschitz filling corollary references herbert abels finite presentability groups compact presentability solvable groups lecture notes mathematics armand borel linear algebraic groups volume springer science business media martin bridson haefliger metric spaces curvature volume springer science business media yves cornulier romain tessera geometric presentations lie groups dehn functions arxiv preprint david epstein mike paterson james cannon derek holt silvio levy william thurston word processing groups peters mikhail gromov geometric group theory vol asymptotic invariants infinite groups shoshichi kobayashi katsumi nomizu foundations differential geometry volume new york alexander lytchak stefan wenger robert young dehn functions extensions asymptotic cones arxiv preprint ayato mitsuishi takao yamaguchi locally lipschitz contractibility alexandrov spaces applications pacific journal mathematics robert young dehn function slpn annals mathematics david bruce cohen department mathematics university chicago university avenue room chicago illinois davidbrucecohen
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abstract anns yielded htm lstm sparsity finally universe space time gravity electromagnetism sequences perceive ranges color pitch vision stimuli occlusion phenomenon perceives senses cells neurons soma dendrites synapses axon connections dendrite dendros tree neurons expectation cortical areas cortical columns macrocolumns minicolumns cells dendrites synapses axon predictive expective sparse cyan magenta predictive connection strength active inactive example active inactive forgotten forests enough active active inactive active inactive overlapsynapses activateneurons learnsynapses grow shrink move predict inactive overlapsynapses predictneurons decodeneurons quack encode encode encode however synapseaddresses value active note synapseaddresses predicted active synapseaddresses active inactive active inactive stimuli forest stimuli forest stimuli forest stimuli forest acknowledgements numenta references mcculloch marblestone bengio silver https hawkins http hawkins hawkins https cui ahmad laukien https https savien
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apr coleman automorphisms finite groups minimal normal subgroups arne van antwerpen abstract paper show coleman automorphisms finite group minimal characteristic subgroup inner therefore normalizer property holds groups using methods show holomorph wreath product finite simple groups among others coleman automorphisms application theorems provide partial answers questions raised hertweck kimmerle furthermore characterize coleman automorphisms extensions finite nilpotent group cyclic lastly note automorphisms order groups inner extending result hai introduction let finite group let denote integral group ring denote center unit group normalizer well known normalizer problem asks whether problem posed question sehgal first results nilpotent groups groups normal sylow due respectively coleman saksonov jackowski marciniak work among others sehgal parmenter jespers hertweck hertweck hertweck kimmerle enlarged class groups positive answer greatly mazur showed question closely related isomorphism problem poses question whether isomorphism two integral group rings implies groups isomorphic hertweck discovered counterexample isomorphism problem first constructing counterexample normalizer problem counterexamples normalizer problem known date obtained using hertweck construction hence opens question whether still large classes groups positive answer normalizer problem say satisfy normalizer property jackowski marciniak realized coleman result normalizer property closely related automorphisms underlying finite group later hertweck kimmerle initiated study coleman automorphisms generalization automorphisms obtained study normalizer property using alternative characterization normalizer problem obvious finite groups coleman automorphisms normalizer property first section give necessary background automorphisms play second section show coleman automorphisms groups minimal characteristic subgroup trivial using techniques show holomorph finite simple groups coleman automorphisms also partially answer questions posed hertweck kimmerle moreover show question answered positively question furthermore theorems proven section show certain products simple groups symmetric groups normal base group coleman automorphisms particular recent results hai guo easy consequences theorems third section coleman automorphisms extensions nilpotent groups cyclic characterized nice consequence provides easier different proof dade result finite abelian group realized group outer coleman automorphisms metabelian group furthermore recent results straightforward applications characterization also shows theorems previous sections naively adapted case product nilpotent group normal base group last section note recent result hai easily generalized shown coleman automorphisms order certain groups inner corollary prove coleman automorphisms order certain groups inner preliminary results section give definition investigated automorphisms also include crucial known results automorphisms aut respectively inn denote automorphisms respectively inner automorphisms group conj denote inner conjugation group outer automorphisms aut set consisting prime divisors order finite group denoted cyclic group order denoted positive integer commutator two elements denoted begin definition coleman automorphism introduced hertweck kimmerle definition let finite group aut prime dividing order sylow exists conj said coleman automorphism group coleman automorphisms denoted autcol image denoted outcol autcol definition let finite group aut conjugacy classes invariant exists called automorphism group automorphisms denoted autc image denoted outc autc following proposition shows prime divisors orders groups automorphisms coleman automorphisms restricted prime divisors well known automorphisms proof found proposition proof coleman automorphisms may found proposition theorem let finite group prime divisors also prime divisors following theorem shows coleman automorphisms behave well respect direct products proposition let finite groups autcol autcol autcol outcol outcol outcol following lemma shows coleman automorphisms close result appeared implicitly lemma let normal subgroup finite group autcol aut proof let autcol let distinct prime divisors order write xpk xpi autcol exist xpi xpi thus xpk therefore proof following lemma found lemma lemma let prime number aut order assume exists normal subgroup idn induces identity induces identity moreover also fixes sylow elementwise conjugation element following result proven hertweck kimmerle theorem let normal subgroup finite group prime number divide order following properties hold aut coleman automorphism order induces coleman automorphism respectively outc outcol outc outcol respectively second part theorem extended follows proof similar included completeness sake theorem let normal subgroup finite group prime number divide order outc outcol outc outcol proof assume outc outcol let coleman automorphism order theorem induces coleman automorphism order hence exists conj put conj let order put differs inner automorphism thus sufficient prove inner hence may assume coleman automorphism order idn theorem induces identity sylow contained follows lemma inner automorphism analogue coleman automorphims introduced gross definition let finite group let aut automorphism prime divisor automorphism centralizes sylow said automorphism note every coleman automorphism group upto inner automorphism automorphism prime theorem gross showed following useful theorem note formulation differs somewhat original theorem let finite simple group let odd prime dividing order every automorphism order inner hertweck kimmerle continued study automorphisms simple groups proved following interesting result theorem finite simple group prime dividing automorphisms inner automorphisms groups characteristic subgroup section coleman automorphisms finite groups minimal characteristic subgroup investigated results groups partially answer several questions posed hertweck kimmerle furthermore results imply several results showing coleman automorphisms certain products particular wreath products coleman automorphisms results recent paper wreath products symmetric groups straightforward applications results moreover shown result gross corollary slightly generalized theorem let normal subgroup finite group let minimal characteristic subgroup every coleman automorphism inner particular normalizer problem holds groups proof note minimal characteristic subgroup necessarily characteristically simple hence direct product finite simple groups isomorphic fixed simple group say abusing notation identify theorem exists prime number automorphisms inner let aut coleman automorphism order first show may assume automorphism let sylow exists sylow denotes image canonical isomorphism put conj conj follows coleman automorphism restricts identity let denote order put coleman automorphism order idp inn may replace hence without loss generality may assume indeed particular idqi note normal subgroup generated precisely lemma aut hence shows aut thus automorphism hence exists conj thus conj proof continues two mutually exclusive cases first consider case put coleman automorphism idn let denote part order put coleman automorphism order idn inn may replace hence without loss generality may assume except may anymore idn characteristic follows hence since follows shows idk desired second consider case abelian thus cyclic group order abelian inner follows idn moreover characteristic follows hence shows idn follows lemma inner assume exists let order xnp implies hence shown order autcol inner hence result follows similar arguments used prove following theorem theorem similar gross corollary theorem let normal finite group automorphisms particular noninner coleman automorphisms proof let prime number let automorphism order hence normal assumptions idp therefore thus assumption consider two cases first suppose exists order shows follows shown idg second assume idp since fixes sylow follows lemma inner automorphism cases shown automorphism prime power order inner coleman automorphism upto inner automorphism result follows immediate consequence theorem get following corollary let finite let finite group let associated semidirect product coleman automorphisms recall wreath product two finite groups defined semidirect product acts left multiplication indices immediate consequence theorem following corollary let finite simple group finite set indices finite group group coleman automorphism thus normalizer problem holds particular wreath product coleman automorphisms straightforward applications theorems mention recent result hai corollary let positive integers coleman automorphisms particular satisfies normalizer property another application reasoning theorems holomorph groups recall finite group holomorph defined hol aut action aut evaluating automorphisms theorem let finite simple group hol coleman automorphisms proof let prime number let autcol hol order theorem exists prime number automorphisms inner may assume order modifying inner automorphism proof theorem taking suitable power lemma aut inn aut hence induces automorphism order inn theorem induces inner automorphism conj conj inn order sylow hence conj conj still fixes conj elementwise induces identity inn thus theorem conj hence conj automorphism inn restricts identity induces identity inn write convenience let wconj wconj conj qwconj conj qwconj hence wconj wconj thus similar argument shows hol chol inn clearly chol inn hence idhol last easy application theorems alternative proof extension recent result hai corollary let finite nilpotent group hol noninner coleman automorphism particular hol coleman automorphisms proof put sylow decomposition hol hol proposition sufficient show result case prime number clearly inn selfcentral normal hol thus theorem hol coleman automorphisms following lemma easily follows fact automorphism group direct product characteristic subgroups direct product automorphism groups lemma let finite simple groups positive integers aut snkn aut aut snkn recall generalized fitting subgroup finite group defined fitting subgroup maximal nilpotent normal subgroup layer maximal semisimple normal subgroup well known elements layer fitting subgroup commute depth results refer section application main results section give partial answers following questions posed hertweck kimmerle prime number finite group outcol chief factor outcol trivial trivial outcol trivial unique minimal normal subgroup hertweck kimmerle showed questions affirmative answers case example solvable recall every finite solvable group abelian minimal normal subgroups concerning third question minimal normal subgroups abelian follows extra assumption minimal normal subgroup abelian partial answer given hertweck kimmerle next theorem show minimal normal subgroup statement always positive answer theorem let finite group unique minimal normal subgroup outcol proof normal subgroup minimality assumption yields obviously assumptions yield minimal characteristic subgroup thus result follows theorem thus answer question groups abelian minimal normal subgroup need checked moreover groups may assumed show positive answer second question implies positive answer third question proposition question positive answer question proof theorem sufficient consider finite groups abelian unique minimal normal subgroup say characteristically simple group hence elementary abelian prime thus unique minimal normal subgroup follows result follows denote maximal normal group following theorem serves provide partial answer question additional assumption theorem let finite group odd prime number suppose normal subgroup direct product simple groups divides order every simple group decomposition outcol proof let coleman automorphism order put snkn lemma aut let sylow qknn reasoning proof theorem may assume without loss generality siki characteristic subgroup snkn thus aut siki let identify copy denoted siki follows assumption clearly normal subgroup generated shows aut since odd follows theorem exists conj holds exists conj hence previous proofs modification inner automorphism taking suitable power may assume without loss generality ide possibly sacrifice thus follows shows thus idk theorem let finite group odd prime number divides order every simple subgroup outcol proof clearly assumptions implies hence shows moreover follows direct product simple groups result follows theorem coleman automorphisms groups section obtain characterization coleman automorphisms finite cyclic groups apart giving concrete examples groups known outer coleman automorphisms obtain easy second proof dade result shows finite abelian group realized group outer coleman automorphisms finite metabelian group furthermore results show results previous section expected products nilpotent group arbitrary finite group faithful action give insight matter proof quite technical first show result product finite abelian group cyclic group prime power order work inspired recent work studied group coleman automorphisms finite generalized dihedral group obtained concrete characterization group first define automorphisms induce outer coleman automorphisms definition let free product finite abelian group cyclic group hxi order prime number write apn sylow decomposition integers define group homomorphisms api xki easily verified maps induce coleman automorphisms possible products theorem let notation definition let smallest positive integer conj xri assume sylow subgroups ordered autcol inn thus zri proof clearly subgroup autcol first study automorphisms determine outer automorphism let zri conj write bxl number api xji equivalently xki shows xki chxi thus mod furthermore idhxi thus hxi calculations show determine outer automorphism differences constant modulo follows elements induce different elements outcol remains show every coleman automorphism induces outer automorphism element let autcol hxi exists conj define automorphism conj clearly coleman automorphism moreover every exists positive integer conj xji hence conj nonnegative integers induce element outcol follows autcol inn application give alternative proof dade result proof makes use well known fact prime number number exist infinitely many prime numbers primitive root unity theorem finite abelian group exists product finite finite abelian groups outcol proof first deal case cpri prime number positive integers define let integer remark exist different prime numbers positive integers whose images zqi primiwrite cps hxi cqi hyi define tive pri root unity product yiki theorem outcol cqi cps second let finite abelian group write hpn decomposition sylow subgroups exist abelian groups positive integers outcol cpsi hpi denote cpsi define group clearly cpsn elements cpsi act trivially except acts like theorem outcol tackle main theorem section proof quite technical crucial steps treated lemmas lemma let prime number let normal subgroup finite group assume nilpotent hxn cpn positive integer suppose sylow suppose autcol idhxi positive integer conj commute furthermore integer proof course result follows nilpotency assume assumption thus hence wxj shows show second statement note sufficient show centralizes let axk thus axk applying definition hxi seperately axk wxj awxj bwxj rewritten wxk wxk equivalent shows lemma let previous let lemma sylow subgroups let kand positive integer let positive integers exists autcol defined via xji proof first check proposed mapping suppose axj bxt positive integers hxi thus exists integer xlp divide hxi thus mod thus need consider case divides without loss generality may assume sylow thus need show axj bxlp xlp bxt image applying relations let decomposition commute find bxlp xlp xjk xjk xlp hand xlp xjk xlp shows relations induce map second show homomorphism let decompositions show quick calculation shows equivalent equality xjk xjk thus find need following equalities rewritten thus assumptions theorem homomorphism clearly coleman automorphism using notation previous lemmas define following subgroup positive integers theorem let prime number cyclic group let nilpotent normal subgroup cpn positive integer let hxi let sylow subgroups let denote smallest positive integer conj xri conj renumber put transversal cpj hxi hhj sylow put transversal cpi hxi autcol inn furthermore outcol proof note clearly subgroup autcol show autcol inn let autcol exists positive integer acts conj wxk sylow containing hxi denote conj idhxi coleman automorphism follows exists positive integer conj put conj clearly positive integers note lemma follows second show element different rest let positive integers suppose conj clearly hxi xji equivalently denoting xji xli put axf sylow decomposition hence hxi thus previous equation equivalent xli hence mod pri sylow exists follows xlj normal subgroup follows xlj equivalently cpj hxi hhj find sylow subgroups cpi hxi remain invariant conjugation follows equivalent cpi hxi shows elements different even inner automorphism every coleman automorphism exist positive integers induce outer automorphism together first graph proof shows autcol inn furthermore shows inn thus autcol inn groups sylow section slight generalization hai recent paper shown using earlier stated results note concept coleman automorphism paper includes among others restriction well completeness sake include statement hai theorem theorem let finite group let nilpotent normal subgroup cyclic sylow abelian every coleman automorphism order inner first show theorem number replaced prime number one easily adapt proof hai general case however first need formulate following corollary lemma corollary let finite group let normal subgroup quotient group abelian let coleman automorphism inner automorphism conj sylow show result one needs modify idn done considering conj taking suitable moreover clear induces identity theorem result follows lemma theorem let finite group let nilpotent normal subgroup cyclic let prime number sylow abelian every coleman automorphism order inner previous result extended groups show general theorem groups reduced general theorem groups theorem let group let nilpotent normal subgroup nilpotent sylow abelian sylow cyclic coleman automorphisms order power proof let statement theorem hence exists hxi sylow put hxi clearly normal subgroup theorem theorem shown outcol outc satisfies conditions theorem clearly holds references blackburn finite groups normal subgroups nontrivial intersection journal algebra coleman modular group ring proc amer math dade locally trivial outer automorphisms finite groups dade correction locally trivial outer automorphisms finite groups gross automorphisms centralize sylow journal algebra hai coleman automorphisms finite nilpotent groups cyclic groups acta mathematica sinica english series hai normalizer property integral group rings holomorphs finite nilpotent groups symmetric groups algebra hai guo normalizer property integral group ring wreath product two symmetric groups hertweck automorphisms finite groups journal algebra hertweck counterexample isomorphism problem integral group rings finite groups annals mathematics pages hertweck local analysis normalizer problem journal pure applied algebra hertweck coleman automorphisms finite groups monatsh hertweck jespers automorphisms normalizer property blackburn groups journal group theory hertweck kimmerle coleman automorphisms finite groups huppert endliche gruppen springer jackowski marciniak group automorphisms inducing identity map cohomology pure appl algebra hai normalizer property integral group rings wreath products finite nilpotent groups cyclic groups communications algebra coleman automorphisms generalized dihedral groups acta mathematica sinica mazur isomorphism problem integral group rings infinite groups mazur normalizer group unit group group ring journal algebra saksonov group rings finite sehgal units integral group rings longman
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bounds bayes risk supervised learning nov matthew ahmad robert wayne state university detroit email mit cambridge email beirami duke university durham email abstract present framework bounding number labeled samples needed train classifier parametric bayesian setting using ideas theory derive bounds average distance learned classifier true maximum posteriori surrogates excess classification error due imperfect learning provide lower upper bounds function using loss distortion measure maximum priori classifier terms differential entropy posterior distribution quantity called interpolation dimension characterizes complexity parametric distribution family addition expressing information content classifier terms lossy compression function also expresses minimum number bits learning machine needs extract training data order learn classifier within specified tolerance use results universal source coding express information content training data terms fisher information parametric family number training samples available result framework computing lower bounds bayes risk framework complements probably approximately correct pac framework provides minimax risk bounds involving dimension rademacher complexity whereas pac framework provides upper bounds risk data distribution proposed framework lower bounds risk averaged data distribution evaluate bounds variety data models including categorical multinomial gaussian models case bounds provably tight orderwise two cases prove bounds tight multiplicative constants index terms supervised learning theory bayesian methods parametric statistics ntroduction central problem statistics machine learning supervised learning learning machine must choose classifier using sequence labeled training samples drawn unknown distribution effectiveness learned classifier measured accuracy classifying future test samples drawn distribution standard approaches problem include support vector machines random forests deep neural networks supervised learning fundamental question sample complexity many training samples necessary learn effective classifier prevailing approach characterizing sample complexity work presented part ieee machine learning signal processing workshop boston ieee symposium information theory barcelona spain jul probably approximately correct pac framework provides almost sure bounds sample complexity families classifiers irrespective data distribution bounds expressed terms dimension captures combinatorially complexity families classifiers typical result goes follows classifier family dimension given training samples excess error probability learned classifier best classifier family high probability pac framework leads empirical risk minimization erm structural risk minimization srm frameworks model selection system designer considers sequence classifier families increasing dimension chooses family minimizes pac bound available training set pac bounds available many popular classifiers including svms neural networks refinements pac bounds provide tighter bounds risk bounds based dimension rademacher complexity developed recently local rademacher averages bounds tighten results cases offering improvement predicted sample complexity bounds one imposes prior set classifiers developed limitation pac bounds characterize minimax performance distribution family may lead pessimistic predictions relative practical peformance indeed authors put way assumption adversary capable choosing worstcase parameter sometimes practice parameter incurs risk may appear small around many researchers studied bounds rissanen proposed minimum description length mdl criterion model selection leverages results universal source coding select complexity model class avoid overfitting mdl framework since seen wide use machine learning see references therein recent survey connections selection also studied resulting akaike bayes information criteria information bottleneck methods paper develop framework computing bounds bayes risk estimating posterior supervised learning particular concerned soft classification performance learning machine rather measure performance strictly terms error probability learned classifier measure well learning machine estimate posterior function turn used classify test samples via map rule quality estimated posterior measures well one classify future test samples also well one estimate confidence level learned classifier time encounters test point develop framework paper bayesian parametric setting joint distribution data points labels belongs known parametric family parameters index distribution drawn known prior example gaussian classification class multivariate gaussian fixed covariance mean taken subvector suitable prior computational purposes conjugate prior case gaussian proposed framework provides lower bounds average distance true posterior posterior estimated samples drawn bounds averaged prior exhibit pessimism minimax bounds risk surrogate excess classification error bounds errors give insight performance learned classifier furthermore approach connects problem learning classifier problem learning distribution fixed posterior merely distribution learn training samples problem learning distribution rich history dating back estimator continuing recent results proposed framework identifies relationship supervised learning lossy source coding parametric bayesian setting posterior distribution function random parameters therefore random object take distance distortion function bound number nats needed describe posterior within specified follows main result paper order drive average error threshold mutual information training samples parameters must least great differential entropy posterior plus penalty term depends quantity called interpolation dimension measures number data points posterior distribution needed uniquely interpolate entire function resulting framework complementary pac framework whereas pac framework considers families classifiers provides generalization bounds hold data distribution framework considers families data distributions provides generalization bounds hold classifier whereas dimension characterizes combinatorial complexity family classifiers interpolation dimension characterizes sample complexity parametric family data distributions larger interpolation dimension training samples needed guarantee classifier performance also emphasize bayes risk lower bounds derived terms generalize usual explicit connection theory learning investigated bounds generalization error derived samples subject lossy compression along similar lines lower bounds distributed learning function network derived contributions paper follows formally laying problem statement section present bounds functions bayes classifiers section iii functions take posterior random source represented risk distortion measure consider two definitions bayes risk one averages risk parameter test point one averages parameter considers worst case test points live subset first definition characterizes average performance overall whereas second definition allows one focus particular region test points bounds function terms object called interpolation set related quantity called interpolation dimension difficult directly analyze information content posterior continuous distribution set uncountably many random vectors address issue define sufficient set points describe posterior called sufficient interpolation set interpolation set finite cardinality one easily bound note information entropy measured nats throughout paper distortion function considering finite samples posterior resulting bounds involve differential entropy posterior evaluated elements interopolation set distortion criterion cardinality interpolation set termed interpolation dimension section translate bounds bounds bounds sample complexity applying bayesian version theorem find mutual information training set parameters random bayesian setup scales log depends determinant fisher information matrix averaged distribution family using fact derive bounds number samples needed ensure small error also discuss challenges opportunities deriving matching bayes risk outer bounds section consider several case studies first consider simple problem estimating discrete distribution samples case derive bounds function distribution lower bounds bayes risk bounds tight asymptote agree almost exactly minimax error consider learning binary multinomial classifier model popular document classification derive bounds function bayes risk provably order optimum carry similar analysis binary gaussian classifiers finally consider simple classification problem case resulting risk falls instead obtained previous cases also show bounds nearly tight give conclusions section notation let denote fields real numbers integers respectively let denote set reals let capital letters denote random variables vectors lowercase letters denote realizations let diag denote matrix elements diagonal let denote expectation subscript indicating random variable expectation taken necessary let denote cardinality set function finite set let denote vector function evaluations points suppressing arguments clarity permits let integer let denote mutual information denote differential entropy use natural logarithm throughout quantities measured nats let denote unit simplex let denote gamma function let denote digamma function log two scalars let denote beta function vector let denote multivariate beta function let denote normal distribution mean variance let beta denote beta distribution shape parameters density function let dir denote dirichlet distribution density function roblem tatement consider problem supervised learning parametric statistical framework let data point label distributed according indexes parametric family distributions alphabet may discrete continuous former case joint probability mass function data point label latter case abuse notation slightly refer joint probability density function even though discrete random variable suppose nature selects learning machine obtains sequence samples denoted pair drawn according learning task select classifier training samples principle classifier may function known one could choose maximum posteriori map classifier minimizes classification error wmap arg max calculated according bayes rule course supervised learning data distribution unknown map classifier unavailable instead suppose learning machine knows parametric family specific distribution objective supervised learning characterize performance learned classifier function number training samples natural performance metric gap misclassification error learned classifier wmap wmap wmap probabilities computed according joint distribution discussed introduction minimax loss respect characterized pac framework family classifiers containing map classifier dimension wmap distribution high probability distribution instead misclassification error gap analyze bayes risk learning posterior analyze soft classification capability learning machine performance averaged distributions indexed let prior distribution parametric family proposed framework presents performance bounds averaged family distributions according view bayes error different ways first represents true distribution parameters space bayes risk simply average loss many instances learning problem second bayes risk represents lower bound minimax risk depending strength prior distribution bayes risk may much smaller furthermore rather study classification error gap study loss define losses terms posterior distribution also called regression function rather choose classifier directly learning machine estimates regression function later used classify test points according map rule underscore point let denote true regression function takes input produces output vector probabilities belongs class also let learning rule maps training samples regression function every loss defined distance vector formed bounds loss rather classification error valuable several reasons first loss related fact see averaged bounds classification loss somewhat less fact relationship holds pointwise see discussion classifier using regression function classification error satisfying point averaged via norm equivalence one derive similar bounds second loss gives comprehensive sense classifier performance classifying signals one wants know probable class confidence one classification small risk ensures classification error also good estimates accuracy classification decision similarly one wants use classifier minimize arbitrary expected type misclassification incurs different one needs accurate estimate regression function finally point relationship divergence omit dependence indicate regression function estimate made ignorance let lkl log divergence evaluated test point pinsker inequality states lkl therefore lower bounds bayes risk translated bounds divergence averaged divergence important place modern machine learning guise loss log loss popular criterion training learning models including deep learning bayes risk learning rule computed taking average loss functions defined three variables take averages data point training set parameterization index illustrative parse impact averaging different random variables end consider two definitions bayes risk first definition termed risk involves averages suppose test point lives set set consider loss compact may consider bayes risk otherwise may beneficial consider performance compact subset consider loss averaged worst case formalize following definition definition define risk learning rule respect loss sup analogous definition holds respect loss bayes risk points may pessimistic especially set large also consider loss averaged term simply bayes risk definition define bayes risk respect loss analogous definition holds respect loss bayes risk simply average performance measured terms loss averaged data distribution data point training set note case normalizing power taken outside expectation basic ingredient results analysis bayes risk essence characterize minimum number nats learning machine must extract training set order obtain representation regression function specified bayes risk tolerance end characterize function posterior learning machine hopes learn suppose compressed lossy version posterior define rate distortion functions respect bayes risk functions definitions definition function regression function respect risk loss rpx inf mutual information true approximated regression function respect bayes risk function inf challenge computing functions defined regression function collection many random many uncountable alphabet mutual information analyzing requires care much section iii given development techniques analysis function interesting right usual sense indeed one learned regression function parametric model one might ask much information required encode posterior transmit another party per separation theorems one needs rpx nats order ensure reconstructed posterior bayes risk respectively therefore following analysis implications distributed learning networks subject taken future work nevertheless main motivation studying function derive lower bounds bayes risk addition quantifying many nats one needs encode regression function bayes risk tolerance function quantifies many nats learning machine needs extract training set order learn regression function tolerance furthermore one quantify maximum number nats one extract training set via mutual information distribution index putting two ideas together one derive necessary conditions bayes risk formalize notion following lemma lemma whenever learning rule risk bayes risk less equal conditions rpx hold respectively see appendix proof intuition behind lemma number nats required learn bayes risk greater given number nats provided training set greater mutual information number nats provided must satisfy number nats required illustrate analogy proposed framework standard theory figure encoder decoder learning machine fig analytical framework paper connection theory rate distortion source distribution gives rise sequence encoded one indices decoder infers index noisy reconstruction average distortion depends encoding rate via function paper prior distribution gives rise data distribution associated regression function treat training samples drawn imperfect encoding regression function learning machine infers noisy estimate estimation error depends number samples iii esults section devoted developing bounds functions extension bayes risk functions defined previous section first define necessary concepts regression function potentially uncountable collection random variables one point mutual information joint entropy uncountably many random variables difficult analyze directly makes difficult compute functions therefore analyze informationtheoretic quantities sampled version acts sufficient statistic entire function capture notion defining interpolation set interpolation dimension definition let finite set let matrix matrix columns evaluations first points regression function points say interpolation set regression function differential entropy finite words interpolation set sampling regression function point regression function respect randomness example section consider binary classification problem regression function logistic form exp recall unknown parameter playing role regression coefficients one chooses set linearly independent vectors straightforward verify joint density random variables exists joint differential entropy finite long however one adds another point first points completely determine regression function value point resulting joint distribution singular joint entropy depending one definitions undefined equal interpolation set provides representation perhaps infinitedimensional regression function even discrete gives compact representation follows inequality learning rule important special case inequality holds equality definition interpolation set said sufficient equivalently interpolation set sufficient roughly speaking interpolation set sufficient one recover entire regression function samples indeed logistic regression example considered set linearly independent points sufficient interpolation set function evaluations one solve exactly recover regression function values cardinality sufficient interpolation set play prominent role analysis definition interpolation dimension denoted cardinality smallest sufficient interpolation set interpolation dimension characterizes number distinct evaluations regression function needed reconstruct sense akin nyquist rate sampling theory expressing many function evaluations takes characterize function known structure interpolation dimension characteristic parametric family indeed measures complexity manner similar dimension whereas dimension characterizes complexity family classifiers many points shatter interpolation dimension characterizes complexity family distributions many sample points regression function needed reconstruct emphasize number function evaluations regression function needed interpolation set distinct number independent samples drawn distribution needed learn regression function learning machine never sees regression function evaluations supervised learning samples drawn source distribution interpolation set dimension tools facilitate analysis functions bayes risk associated parametric family nevertheless show although distinct concepts number training samples needed learn regression function related interpolation dimension presenting bounds need final technical condition parametric family definition interpolation set said onto every matrix set least one qij qij words interpolation set defines mapping onto mapping onto set set valid probability vectors truncated first elements interpolation set onto realize valid probability vector choosing examples consider paper interpolation sets onto however one define parametric models regression function takes subset possible probability vectors consider trivial example bernoulli parametric family case set regression functions restricted onto interpolation set exists bounds involving first result bound function rpx terms interpolation dimension entropy regression function theorem let interpolation dimension suppose exists onto sufficient interpolation set cardinality let interpolation set necessarily sufficient onto function rpx bounded log log log log min proof provided appendix content theorem rate distortion function least great differential entropy regression function evaluated interpolation set less penalty term involves expected distortion cardinality interpolation set higher cardinality interpolation dimension larger function nats needed describe regression function average emphasize lower bound holds interpolation set cardinality less interpolation dimension one simply obtains looser bound rpx however upper bound depends interpolation dimension fact invariant choice interpolation set sense interpolation dimension fundamental quantity figuring prominently upper lower bounds function corollary assumptions theorem function rpx bounded log log log bounds involving bounds presented theorem set test points single poorperforming point forces high value risk performance useful metric also interested knowing performance test points bound performance carry analysis mutual information averaged ensemble interpolation sets requires additional machinery definition let index set either countable uncountable say function interpolation map every interpolation set words interpolation map defines collection disjoint interpolation sets elements interpolation sets need sufficient interpolation sets size interpolation set need interpolation dimension every found interpolation set see interpolation map covers useful analysis definition let range let probability integral becomes sum countable find convenient work interpolation maps isotropic respect probability distribution definition say interpolation map isotropic every words interpolation map isotropic points interpolation set lie level sets probability distribution one define isotropic interpolation map one bound bayes risk expressions similar theorem theorem suppose either countable uncountable density riemann integrable suppose also exists onto sufficient interpolation set cardinality let isotropic interpolation map dimension probability rate distortion function bounded log log log log min expectation taken distribution denoting point inverse image furthermore lower bound holds interpolation map regardless whether isotropic see appendix proof content theorem similar bounds rpx function depends entropy interpolation set penalty term associated permissible risk however entropy averaged interpolation map allows account average error points range interpolation map whereas previous result accounted error subset corollary assumptions theorem rate distortion function lower bounded log log log ample omplexity ounds combining lemma theorems one compute lower bounds bayes risk turn sample complexity lower bounds bounds incorporate interpolation dimension differential entropy posterior mutual information training samples distribution index end one must evaluate two quantities mutual information training set parameterization differential entropy regression function evaluated points interpolation set perhaps averaged interpolation map appropriate conditions quantities expressed simpler terms permitting explicit computation sample complexity bounds present expressions respectively derive sample complexity bounds expression smooth posterior using results universal source coding express terms differential entropy fisher information matrix distribution family let minimal sufficient statistic meaning function sufficient statistic let let denote fisher information matrix respect log fisher information roughly quantifies amount information average training sample conveys appropriate regulatory conditions make notion precise bound mutual information terms number training samples theorem let parametric family minimal sufficient statistic suppose fisher information matrix exists finite determinant suppose estimator asymptotically efficient converges normal distribution zero mean covariance matrix following expression holds log log proof follows celebrated theorem see averaging bounds derived clarke barron yields result upshot information conveyed training set grows log times effective dimension parameter space constants determined sensitivity distribution parameters quantified fisher information matrix prior uncertainty parameters expressed expression intuitive light assumption central limit theorem holds estimator approaches gaussian distribution limit increasing resulting mutual information includes term associated differential entropy however result holds smooth distributions fisher information matrix also section consider case smoothness assumption hold changes scaling expressions differential entropy unusual quantity compute must evaluate density posterior distribution evaluated finitely many points take expected logarithm density distribution often difficult evaluate closed form evaluating expected logarithm difficult still therefore expect expression always available problems interest nevertheless develop intuition consider interpolation set example binary gaussian classifier interpolation set basis evaluated indeed common case set vectors form basis case differential entropy following expression theorem define random variables niy every every preceding assumptions differential entropy regression function log log exists niy matrix random variables niy supposing proof given appendix make remarks preceding expression first function parametric family prior resulting distribution simple one compute relatively easily otherwise one may need resort numerical techniques whether simpler estimate numerically terms expression directly estimate depends specific distributions question second continuous distributions first two terms converge probability one log finally even expression difficult compute establishes scaling laws showing increases linearly parametric families satisfying preceding conditions corresponding continuous distributions also bound entropy theorem interpolation set satisfying conditions stated posterior entropy satisfies log see appendix proof fortiori expression bounds expected entropy interpolation map entropy grows linearly interpolation dimension see entropy decays least fast log bound necessarily tight useful estimation bounds true entropy difficult obtain example gaussian case considered section find substituting preceding bound place exact differential entropy negligibly impacts resulting bounds finally interpolation map identified expected entropy computed numerically via monte carlo methods presented long one sample easily one produce arbitrarily many samples estimate entropy one average appropriate interpolation map emphasize computation need carried given distribution model risk bounds evaluated sample complexity bounds theorem derive explicit sample complexity bounds bayes risk substituting lemma lower bound theorem obtain necessary condition log log log log log interpolation set permissible loss similar bounds hold bayes risk using theorem describe intuitively terms complicated expression expression involving fisher information matrix describes many samples needed average learn minimal sufficient statistic thus however objective learn sufficient perhaps necessary learn order learn second term captures notion difference entropies bigger term corrects number samples needed following term slack term associated bayes risk final term arises approximation impose regularity conditions eigenvalues differential entropies derive following bounds sample complexity proposition addition conditions theorem suppose log positive constants exp proof algebraic manipulation theorem special case dimensionality minimal sufficient statistic equal interpolation dimension times alphabet size obtain rule agrees known scaling sample complexity case one derive pac framework case necessary sufficient number training samples grow quadratically required precision emphasize scaling particular choice loss assumption smoothness indeed classification loss divergence lkl exist learning problems one derive pac error bounds decay furthermore exist classification problems one derive error bounds even loss important albeit extreme example error free learning problem exists classifier perfectly classifies test samples section derive bounds simple setting resulting bounds imply scaling loss expected achievability natural question whether matching upper bounds bayes risk hold random coding joint typicality arguments prove achievability bounds rate distortion apply supervised learning setting joint typicality arguments depend repeated draws source distribution analogous situation supervised learning would encode multiple posteriors associated different draw course consider single draw furthermore random coding arguments presuppose design control source code would correspond design control distribution labeled data control arguments apply stated proposition mild assumptions one derive lower bounds error scale one apply pac framework derive upper bounds scale case bounds order optimum however constants higher order terms pac bounds differ significantly primarily pac bounds distribution agnostic bayes risk bounds take prior distribution account finally one derive asymptotic achievability bounds bayes risk via analysis estimate posterior suppose posterior lipschitz continuous norm respect conditions theorem hold central limit theorem implies estimation error variance scaling depending fisher information matrix lipschitz assumption bayes risk scales constants terms depending fisher information matrix lipschitz constant ase tudies categorical distribution consider first comparatively simple learning problem learning discrete probability distribution samples drawn distribution mentioned introduction problem studied extensively model problem framework posing learning problem alphabet trivial let let number values random variable may take let family distributions indexed unit simplex posterior distribution simply distribution categorical random variable probabilities indicated compute bayes risk suppose dirichlet prior hyperparameter particular uniformly distributed unit simplex define trivially single sufficient interpolation set interpolation dimension categorical distribution therefore interpolation map trivial since singleton rpx hence use refer entropy dirichlet distribution evaluate differential entropy posterior unique interpolation set log multivariate beta function digamma function allows compute functions theorem categorical distribution rate distortion function bounded log log log log log min proof evaluating bounds theorem differential entropy yields result particular interested lower bounds given corollary categorical distribution rate distortion function lower bounded log log lower bound functions depend cardinality alphabet prior uniform case differential entropy posterior simplifies log log case upper lower bound differ term magnitude grows without bound using lemma translate bounds lower bounds bayes risk requires evaluation mutual information mutual information difficult compute closed form apply approximation theorem lemma learning categorical distribution mutual information log proof appears appendix results imply following bound bayes risk theorem learning categorical distribution bayes risk bounded exp exp exp proof manipulation theorem lemma words bayes risk decays faster constant depends prior distribution unspecified term furthermore derive lower bound minimax error choosing element constant using fact exp obtain lim exp comparison gives upper lower bounds minimax error lower bounds result bounds bayes risk particular shows symmetric priors large lower upper bounds agree suggesting furthermore constant quite close constant sample complexity first term bounds numerically somewhat remarkable proposed method leads bound close actual minimax bound given adapted optimized problem hand open question whether optimization prior eliminate gap point advantages bounds proven theorem first lower bounds theorem hold prior lower bound applicable furthermore negative terms lead negative bound risk many values hand bounds asymptotic unspecified multiplicative constant owing approximation mutual information therefore emphasize objective exercise derive bounds outperform tighter literature illustrate effectiveness proposed framework relatively little effort framework provides risk bounds tight nearly tight within small multiplicative constant multinomial classifier consider binary classification multinomial model model derive bayes risk bounds error bayes risk test points multinomial model alphabet suppose observation distribution known constant distribution supported set vectors multinomial distribution generalization binomial distribution given variates drawn categorical distribution counts number times variate value multinomial model ubiquitous text document classification word counts used identify document types essentially class modeled different categorical distribution task classifier determine categorical distribution test samples drawn let denote categorical distribution parameters distribution suppose also uniform priors resulting posterior similar previous subsection suppose dirichlet prior conjugate prior specifip cally let dir also choose let choice allows straightforward derivation function accompanying sample complexity bounds choices posterior form rixi parameterization cost generality terms posterior functions posterior form put form thus distribution induces prior multinomial bayes classifiers bayes risk bounds prior lower bounds minimax risk compute function find interpolation set posterior straightforward see sufficient interpolation set evaluating posterior point kei yields kei rik allows one recover ratio therefore parameter since categorical distribution lies unit simplex one recover entire parameter vector also straightforward see smaller interpolation set sufficient hence using interpolation set bound function data point first need compute requires following lemma lemma scalar function exists log let random variable log log log see appendix proof bound lemma multinomial classification problem described interpolation set differential entropy upper bounded log log log proof appears appendix result state bounds function multinomial classifier theorem binary multinomial classification functions bounded log log log rpx log min note bound distortion test points order bound distortion averaged would need construct interpolation map spans sufficiently large portion alphabet involves constraint difficult construct interpolation map therefore let bound rpx suffice relegate bounds future work finally bounds derive bayes risk bounds difficult compute directly resort approximation theorem lemma multinomial classification problem described mutual information log log log putting results together yields bound risk brevity state bounds easy infer bounds analogy theorem multinomial binary classification problem risk learning rule bounded exp proof manipulation theorem lemma resulting bounds somewhat involved one glean intuition increases family classifiers becomes rich bayes risk increases however error decays mentioned earlier paper dimension bounds imply risk decay slowly lower bound establishes scaling bayes risk however prove tightness constants could categorical case binary gaussian classifier next consider binary gaussian setting let let parameterize data distributions densities gaussian antipodal means known variance choose prior map classifier problem hyperplane passing origin normal model regression function exp model analogue multinomial case posterior form logistic function model however derive bounds function averaged test points see first observe orthogonal basis sufficient interpolation set basis set function evaluation allows one recover inner product xti independent inner products allows one recover interpolate entire regression function furthermore choosing orthogonal basis guarantees elements statistically independent next define interpolation map let index set positive orthant let interpolation map orthogonal basis kxk kvk straightforward verify one choose basis sets disjoint different choices also straightforward verify range interpolation map probability interpolation map order bound function compute expected entropy regression function averaged interpolation map lemma expected differential entropy posterior evaluated interpolation map bounded log log gamma function digamma function defined lower bound divided let denote preceding bound divided bound function brevity state result theorem binary gaussian classification function bounded log log log log min proof evaluation bounds theorem differential entropy lemma furthermore bound bayes risk state results risk brevity theorem binary classification problem bayes risk bounded exp proof first bound first one verify minimal sufficient statistic also immediate log combining facts theorem obtain log however case evaluate mutual information closed form prior gaussian posterior asymptotically gaussian also gaussian let simple calculation shows log discrepancy estimated exact mutual information negligible unless applying bound theorem yields result case obtain scaling law one invoke pac bounds based dimension see scaling also upper bounds bayes risk therefore bound although tightness strict definitions remains open question classification finally present example binary classification problem experiment samples generated truly linearly separable well known required sample complexity drastically changes merely opposed usual case noisy samples see references therein consider parametric distribution single parameter parameter vector let let let let defined follows note joint distribution defined satisfy clarke barron smoothness condition clearly continuous hence need calculate mutual information directly need calculate show fact mutual information follows different scaling case leading different sample complexity scaling law expected compute first focus otherwise otherwise max min note sample exists define similarly sample exists define using bayes rule get otherwise thus log log hence integrating log order statistics note due linearity expectation symmetry written log log harmonic number defined hence evident asymptotically log words scales log contrast rest case studies clarke barron would apply mutual information samples unknown parameter would scale log rather constructing interpolation map case straightforward bound function directly first note straightforward show procedure minimizes estimate hence without loss generality focus strategies hence function calculated maxp maximized pdp value hence given log log log hence combining sample complexity lower bound arrive example fundamentally different scaling opposed previous examples note bound tight consistent matching upper bounds let enumerate bound analyze tightness sample complexity bounds using framework considering bayes risk consider following estimator defined thus due uniform prior hence satisfy risk would suffice hand bounds suggest following lower bound log turn equivalent following euler constant suggested upper bound given scheme lower bound tight within multiplicative factor onclusion presented framework analyzing learnability bayes classifiers supervised learning treating regression function random object derived bounds ratedistortion function average loss showed rate distortion function bounded expressions involving interpolation dimension new quantity characterizes fashion complexity parametric family addition characterizing amount information needed describe bayes classifier specified loss showed ratedistortion function permits derivation lower bounds bayes risk terms number samples evaluated bounds several statistical models cases showing bounds nearly tight important future application work distributed learning bounds characterize much information required learn classifier much information required describe classifier one already learned result expect prove useful characterizing much information network classifiers connected links need exchange order learn collaboratively bayes classifier distributed samples ppendix roof heorem proof theorem involves maximization entropy posterior estimation error subject constraint prove lemma lemma let interpolation set cardinality satisfies constraint differential entropy satisfies log log log proof start bounding holds equality independent hypothesis norm greater every expectation remaining random variables leads optimization problem maximize subject preceding formulation suggests impact objective function therefore clearly optimal set probability one resulting optimization problem maximize subject convex program writing lagrangian get optimizer independent identically distributed set expectation constraint satisfied claim lemma established calculation distribution ready prove theorem proof theorem lower bound inequality inf recall applying lemma yields lower bound upper bound definition rpx sufficient interpolation set inf inf inf inf inf inequality follows posterior function element member thus joint differential entropy bounded zero finally choose jointly independent choose distribution elements following distribution given follows constraint set ensure turn ensure expectation constraint note actual value play role differential entropy merely translation hence value depend upper bound achieved evaluating differential entropy choice ppendix roof heorem lower bound inequality fact supremum dominates average sup inf inf expectation distribution set mass function countable density function uncountable specify presently given subject expectation constraint objective maximize differential entropy term shown proof theorem optimizing distribution given set function log sup defined log log remains take supremum sum logarithm terms respecting distortion constraint simplify optimization problem suppose countable thus probability mass functions uncountable result follows taking limit countable partitions note term show rhs similar proof lemma optimum set follows concavity log optimal constant every clearly optimal set constraint becomes recalling probability range substituting yields log shown upper bound upper bound follows argument bound rpx ppendix roofs heorems proof theorem bayes rule posterior pair interpolation set niy niy order compute need compute density standard result density function mapping samples terms niy jacobian matrix wim nim normalization constraint random variables wiy overdetermined random variables niy therefore without loss generality take nim every straightforward show mapping wiy taking derivatives determinant jacobian denoted let matrix vectors matrix vectors clearly jacobian block diagonal thus jacobian entire mapping denoted density matrix thus density matrix resulting entropy therefore log log log next prove theorem proof theorem proof follows structure proof lemma entropy bounded sum individual entropies furthermore normalized sum regression function fixed must equal one entropy maximized letting random variable uniformly distributed across result follows ppendix roofs emmas proof lemma suppose learning rule satisfies bayes risk constraint lpx respectively let joint probability distribution distribution index training set true posterior learned posterior function training set markov chain therefore repeated applications inequality yield latter mutual information computed according obtained marginalizing joint distribution furthermore take infimum distributions satisify bayes risk constraint inf rhs preceding definition function appropriate bayes risk proof lemma observe overdetermines distribution minimal sufficient statistic minimal sufficient straightforward calculation demonstrates fisher information matrix diag therefore log log latter equality holds marginal components dirichlet distribution follow beta distribution beta expected logarithm beta random variable log substituting preceding theorem yields result proof lemma straightforward verify invertible jacobian method density differential entropy bounded log log log log log log log log log log log log log log log max log log log log log kei lemma need compute proof lemma since beta distribution second kind density fri therefore differential entropy log log log log log log log log log log log final equalities due expression logarithm random variable similarly log log log remains calculate log log max log log max log log invoking lemma combining terms yields result proof lemma first observe minimal sufficient statistic joint distribution straightforward show fisher information matrix diag theorem mutual information log log log log log desired result proof lemma interpolation set let kxk let denote posterior evaluated ith element set determined interpolation map without loss generality suppose let straightforward computation shows exp first objective find density using jacobian formula pni log mapping jacobian equal therefore exp log next differential entropy observe log log log log exp exp exp therefore log log exp log log exp log log log log follows fact furthermore follows fact exactly half expectation folded gaussian well known bound tight within constant gap log final step take expectation white gaussian distribution perelement variance define random variables yields log log definition order evaluate preceding expectation need compute mean random variable expected logarithm random variable quantities applying yields log log gamma function digamma function eferences vapnik support vector method function estimation nonlinear modeling springer nature statistical learning theory springer muller mika ratsch tsuda scholkopf introduction learning algorithms neural networks ieee transactions vol breiman random forests machine learning vol hinton osindero teh fast learning algorithm deep belief nets neural computation vol bengio lamblin popovici larochelle greedy training deep networks advances neural information processing systems vol vapnik chervonenkis uniform convergence relative frequencies events probabilities theory probability applications vol vapnik overview statistical learning theory ieee transactions neural networks vol sep bartlett williamson anthony structural risk minimization hierarchies information theory ieee transactions vol barron approximation estimation bounds artificial neural networks machine learning vol 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sorting recurrent comparison barbara geissmann stefano leucci liu paolo penna department computer science eth switzerland name surname sep abstract present sorting algorithm case recurrent random comparison errors algorithm essentially achieves simultaneously good properties previous algorithms sorting distinct elements model particular runs time maximum dislocation elements output log total dislocation guarantees best possible since prove even randomized algorithms achieve log maximum dislocation high probability total dislocation expectation regardless running time acm subject classification sorting searching keywords phrases sorting recurrent comparison error maximum total dislocation introdution study problem sorting distinct elements recurrent random comparison errors classical model comparison wrong fixed probability correct probability probability errors independent possible pairs elements errors recurrent comparison repeated time computation result always always wrong always correct scenario sorting algorithms perform equally well terms output likely produce nearly sorted sequences others measure quality output permutation terms sortedness common way consider dislocation element difference position permutation true rank among elements two criteria based dislocations elements total dislocation permutation sum dislocations elements maximum dislocation element permutation naturally running time remains important criteria evaluating sorting algorithms best knowledge recurrent random comparison errors best results respect running time maximum total dislocation achieved following two different algorithms braverman mossel give algorithm guarantees maximum dislocation log total dislocation high probability main drawback algorithm seems running time constant exponent rather high klein give much faster algorithm guarantees maximum dislocation log high probability however provide upper bound total dislocation previous result obviously log research supported snsf swiss national science foundation project number barbara geissmann stefano leucci liu paolo penna licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany sorting recurrent comparison errors steps maximal dislocation total dislocation braverman mossel log klein log log log exp table previous results constant depends error probability success probability algorithm example success probability analysis yields total dislocation follows trivially maximum dislocation analysis given paper investigate whether possible guarantee bounds together algorithm running time maximum dislocation log total dislocation contribution propose new algorithm whose performance guarantees essentially best two previous algorithms see table indeed algorithm window sort takes time guarantees maximum dislocation log probability expected total dislocation main idea iteratively sort elements comparing element neighbouring elements lying within window halve window size every iteration iteration element assigned rank based local comparisons placed according computed ranks algorithm inspired klein algorithm distributes elements buckets according computed ranks compares element elements neighboring buckets obtain new rank halves range bucket iteratively note however two algorithms operate different way essentially following key difference bucket window number elements bucket fixed since computed rank several elements could window instead number elements fixed property essential analysis total dislocation window sort introduces potential offset computed rank computed position element analysis consists showing offset sufficiently small considering number delicate conditions algorithm maintain throughout execution sufficiently high probability first describe standard version algorithm achieves afore mentioned bounds error probability improve result using idea shrinking window size different rate experimental results see appendix show performance standard version significantly better theoretical guarantees particular experiments suggest expected total dislocation maximum dislocation log addition prove sorting algorithm guarantee maximum dislocation log high probability sorting algorithm guarantee expected total dislocation related work sorting comparison errors computing errors often considered framework game called game responder thinks object search space questioner geissmann leucci liu penna find asking questions responder provides answers however answers incorrect purpose responder adversarial lier games extensively studied past various kinds search spaces questions errors see pelc survey cicalese monograph feige studied several comparison based algorithms independent random errors error probability comparison less half repetitions comparison obtain different outcomes comparisons independent required reported solution correct probability proved sorting log comparisons suffice gives also running time model sorting random swaps represented markovian processes studied question number inversions reversed pairs within constant factor total dislocation karp kleinberg studied noisy version classic binary search problem elements compared directly instead element associated coin unknown probability observing heads tossing probabilities increase going sorted order recurring errors coppersmith rurda studied simple algorithm gives weighted feedback arc set tournaments fast problem weights satisfy probability constraints algorithm consists ordering elements based computed ranks unweighted fast identical computed ranks alonso hadjicostas lakshamanan studied quicksort recursive mergesort respectively random comparison errors paper organization rest paper organized follows present window sort algorithm section analyze maximum total dislocation section section respectively explain modify window sort allow larger error probabilities section show experimental results appendix additionally prove lower bounds maximum average dislocation sorting algorithm section window sort window sort consists multiple iterations procedure starting permutation window size compare element left right adjacent elements exist count wins number times comparison outputs larger element obtain computed rank element based original position wins denotes original position computed rank equals max plus number wins get new permutation placing elements ordered computed ranks finally set start new iteration window size first iteration formalize window sort algorithm following assume sort elements refer element rank let denote permutation elements beginning current iteration window sort let denote permutation obtained iteration permutation next iteration performs similarly let denote window size current next iteration furthermore let denote sorted permutation define four important terms element position position sorting recurrent comparison errors algorithm window sort permutation elements initialization initial window size element two variables wins set zero repeat foreach position foreach whose position wins wins max wins place elements ordered break ties arbitrarily set wins zero computed rank current position minus zero negative plus number wins computed position position theorem window sort takes time proof consider three steps algorithm number comparisons iteration outer loop current size window therefore first step needs time second step could apply instance counting sort see takes time since computed ranks lie zero thus total running time upper bounded preliminaries first introduce condition errors comparisons element fixed subset elements depends window size definition define errors set errors among comparisons every theorem window sort returns sequence maximum dislocation whenever initial comparisons hold elements proof theorem follows end section analysis shall prove following computed rank element close true rank dislocation elements small lemma computed rank element indeed close true rank number errors involving element consideration small lemma number positions element move iterations small lemma geissmann leucci liu penna introduce condition implies theorem throughout execution window sort would like every element satisfy following condition window size dislocation also introduce two conditions essentially relax requirement elements satisfy first condition justifies first step algorithm second condition restricts range elements get compared iteration window size element larger smaller elements lying apart positions left right original position window size left right adjacent elements satisfy condition note holds elements also hold elements elements satisfy computed rank close true rank errors comparisons lemma every window size element satisfies satisfy absolute difference computed rank true rank bounded proof follows immediately condition consider difference computed rank computed position element define offset element afterwards consider difference original position computed position element fact observe step algorithm holds every permutation every window size every element difference computed rank lemma permutation elements element difference computed rank every element difference computed position every element proof lemma analogue proof lemma lemma every permutation window size offset every element proof let computed rank computed position larger number elements computed rank smaller number elements computed ranks fact every element computed rank smaller every element computed rank larger lemma consider generic iteration algorithm permutation window size iteration position element changes moreover position element changes algorithm terminates sorting recurrent comparison errors proof fact lemma triangle inequality since halved every iteration final difference finally conclude theorem show dislocation element small number errors small proof theorem consider iteration algorithm current window size show holds elements current iteration implies also holds elements next iteration window size becomes order hold next iteration computed position element differ true rank lemma sufficient require computed rank element differs true rank lemma inequality follows hypothesis thus shown iteration window size elements dislocation lemma subsequent iterations move element positions remark care maximum dislocation could obtain better bound simply stopping algorithm iteration window size guarantees condition high probability order bound also total dislocation let algorithm continue way window size allow show total dislocation linear expectation maximum dislocation section give bound maximum dislocation element running window sort elements prove function probability single comparison fails main result following theorem set elements probability maximum dislocation running window sort log enough prove condition theorem holds log probability least lemma every fixed element every fixed window size log probability geissmann leucci liu penna union bound probability holds condition theorem holds probability least log implies theorem proof lemma since comparison fails probability independently comparisons probability event happens equal probability least errors occur comparisons denote probability show make use following standard chernoff bounds see instance theorem chernoff bounds let independent poisson trials let following bounds hold iii lemma probability least errors occur comparisons satisfies proof let random variable denote number errors outcome comparisons clearly let theorem case similarly theorem case theorem case iii lemma log theorem proof show first case two similar lemma log sorting recurrent comparison errors total dislocation section prove window sort orders elements total dislocation linear times factor depends theorem set elements expected total dislocation running window sort log key idea show element elements adjacent true rank matter upcoming iterations holds sufficient keep following weak invariant element throughout iterations invariant consists three conditions satisfied satisfies condition elements original position satisfy condition elements original position satisfy condition note satisfies elements lying satisfy rest section structured follows first derive several properties weak invariant prove log log bound expected total dislocation finally extend proof achieve claimed linear bound properties weak invariant start key property weak invariant element lemma let permutation elements window size iteration window sort weak invariant holds element computed rank every element differs still holds permutation next iteration window size proof consider set elements computed ranks differ original positions thus elements set elements whose original positions assumption lemma element using reasoning proof lemma conclude element consider set elements lemma computed ranks lie thus thus second condition holds next iteration continue third condition consider set elements assumptions lemma sufficient show every element larger smaller elements whose computed positions smaller larger rest follows second condition show former case latter symmetric distinguish three subcases elements smaller iii case follows immediately third condition case follows immediately second condition geissmann leucci liu penna iii assumption lemma thus computed rank element smaller otherwise thus since assume lemma thus implies first condition still satisfied next iteration concludes proof next adopt lemma analyze probability keeping weak invariant element arbitrary window size several iteration window sort lemma consider iteration window sort permutation elements window size log defined theorem weak invariant element holds probability least still holds window size log iterations window sort proof lemma probability fails next iteration let log logw probability fails iterations window size window size log log log first inequality fact increases decreases lemma log leading statement double logarithmic factor main idea given window sort guarantees maximum dislocation log probability least theorem trivially implies expected total dislocation log precisely expected dislocation log log since fraction elements dislocated others dislocated log next describe improve log log considering analysis two phases execution algorithm phase first phase consists iterations window size log probability least elements satisfy phase phase second phase consists executions window size log elements satisfied end previous phase probability fixed element violates second phase precisely theorem proof theorem probability holds elements window size log least thus restart analysis log corresponding permutation assume element satisfies lemma probability fails window size log lemma element moves positions original position apart true rank therefore expected dislocation element log log log equality holds sufficiently large log sorting recurrent comparison errors linear dislocation proof theorem section apply simple idea decrease upper bound expected total dislocation running window sort elements log recurse analysis previous section several phases roughly speaking iteration window sort halves window size phase iterations logarithmizes window size phase iterations window size log phase subsequent iterations window size log log phase subsequent iterations window size log log log bound expected dislocation element let denote window size phase log log log log log phase would consist single iteration remaining section consider phases equation true call valid phases lemma weak invariant holds window size probability still holds window size least similarly analysis section get valid phase contributes expected dislocation stop analysis valid phases lemma expected dislocation element inequality holds since next define phase valid depends term wlog holds every valid phase decreases increasing instance log log log log log log log therefore choose log use upper bound log log log log log equations lemma imply following lemma expected dislocation element running window sort log immediately implies theorem geissmann leucci liu penna extension reason require error probability smaller analyze probability errors occur comparisons bound number errors appears since halve window size every iteration let window size shrink another rate limit also change first running time adapted window sort become second permutation window size order maintain condition element computed position differ thus differ new issue thus probability errors occur comparisons since expected number errors reasoning lemma thus note change accordingly finally number windows weak invariant also change accordingly let number windows matter weak invariant according analysis section implying course constant inside linear expected total dislocation also change accordingly theorem error probability modified window sort elements takes time maximum dislocation log probability expected total dislocation log lower bound maximum dislocation section prove lower bound maximum average dislocation achieved sorting algorithm let set elements sorted think instance sorting problem pair permutation represents order element input collection matrix encodes result comparisons seen algorithm precisely reported smaller comparing otherwise notice iff hence follows define following lemma whose proof moved appendix key ingredient lower bounds lemma let let possibly randomized algorithm random instance probability returns permutation elements appear wrong relative order least first consequence previous lemma obtain following sorting recurrent comparison errors theorem possibly randomized algorithm achieve maximum dislocation log high probability proof lemma algorithm random instance must return permutation elements appear wrong order probability larger happens least one following two conditions holds position element least position element case maximum dislocation must least log finally also able prove lower bound total dislocation whose proof given appendix theorem possibly randomized algorithm achieve expected total dislocation references alonso chassaing gillet janson reingold schott quicksort unreliable comparisons probabilistic analysis combinatorics probability computing braverman mossel noisy sorting without resampling proceedings annual symposium discrete algorithms pages cicalese search algorithms reliable computation unreliable information monographs theoretical computer science eatcs series springer coppersmith fleischer rurda ordering weighted number wins gives good ranking weighted tournaments acm trans algorithms july cormen leiserson rivest stein introduction algorithms mit press feige raghavan peleg upfal computing noisy information siam journal computing geissmann lengler sorting swaps noisy comparisons appear proceedings genetic evolutionary computation conference gecco geissmann penna sort well comparisons arxiv hadjicostas lakshmanan recursive merge sort erroneous comparisons discrete applied mathematics karp kleinberg noisy binary search applications proceedings eighteenth annual symposium discrete algorithms soda new orleans louisiana usa january pages klein penninger sohler woodruff tolerant algorithms proceedings annual european symposium algorithm pages mitzenmacher upfal probability computing randomized algorithms probabilistic analysis cambridge university press pelc searching games errors fifty years coping liars theor comput persi diaconis spearman footrule measure disarray journal royal statistical society series methodological geissmann leucci liu penna experimental results section discuss experimental results performance window sort increasing error probability see tables results suggest practice probability error much higher theoretical analysis experiments measure average dislocation gives estimate expected total dislocation maximum dislocation among elements note experiments window sort performs significantly better theoretical guarantees see theorem corresponding values also experiments suggest expected total dislocation since average dislocation seems increase cases analogously maximum dislocation seems log consider five different values ten different values error probability setting consists instances use error probability generate comparison table among elements simulating recurrent comparison errors moreover set standard version window sort table table average dislocation maximum dislocation one element log divided omitted proofs proof lemma prove running time algorithm compute within steps sequence correct relative order probability larger first let focus deterministic version algorithm fixing sequence random bits used algorithm call resulting algorithm let denote probability generating sequence random bits sorting recurrent comparison errors random bits come fair coin notice might already deterministic algorithm case every consider instance let permutation elements returned element element sequence returned define new instance swapping elements along results comparisons formally define define comparison matrix accordingly letting random variable space comparison matrices get used fact probability distribution let random variable whose value chosen permutations let probability instance solved notice follows discussion let indicator random variable iff algorithm instance either terminates within steps terminates returning sequence appears let set possible instances set instances appear correct order output notice bijection instances corresponding instances notice use less random bits due limit running time geissmann leucci liu penna transformation injective moreover since indistinguishable either terminate within steps instances consequence either must implies let resp indicator random variable iff either execution resp random instance terminate within steps terminates returning sequence appear wrong relative order calculations know ready bound probability indeed last equality follows fact probability distribution elements hence proof theorem let algorithm let random instance even number elements define indicator random variable iff returns permutation elements elements appear wrong order lemma know hence obtain lower bound expected total dislocation achieved follows
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oct extensions rings driss brahim abdulaziz department mathematics university iowa iowa city usa department mathematics faculty sciences mohammed university rabat morocco driss bennis abdulazizsha abstract notion trivial extension ring module extensively studied used ring theory well various areas research like cohomology theory representation theory category theory homological algebra paper extend classical ring construction associating ring ring family given integer call new ring construction extension particular classical trivial extension extension thus generalize several known results classical trivial extension setting extensions give new ones various constructions properties extensions studied detailed investigation graded aspect extensions also given end paper investigation various divisibily properties extensions context several open questions arise mathematics subject classification primary secondary key words trivial extension extension graded rings homogeneous ideal anderson introduction except brief excursion section rings considered paper assumed commutative identity particular denotes ring modules assumed unitary left modules course commutative ring actually rbimodules let denotes set integers natural numbers set denoted ring residues modulo integer noted recall trivial extension ring denoted whose underlying additive group multiplication given since introduction nagata trivial extension rings also called idealization since reduces questions modules ideals used many authors various contexts order produce examples rings satisfying preassigned conditions see instance known trivial extension related following two ring constructions see instance section generalized triangular matrix ring let family rings family modules bimodule assume every exists homomorphism denoted multiplicatively every set consisting matrices extensions rings usual matrix addition multiplication ring called generalized formal triangular matrix ring denoted also see trivial extension naturally isomorphic subring consisting matrices note since commutative symmetric algebra recall symmetric algebra associated graded ring quotient graded tensor trn homogeneous ideal generated note graded general image naturally isomorphic graded also worth recalling free basis trivial extension also naturally isomorphic set indeterminates particular inspired facts introduce extension classical trivial extension rings extensions associated modules integer literature particular cases extensions used solve open questions authors introduced extension used give counterexample faith conjecture also case extension introduced give example ring injective hull compatible multiplication gave negative answer question posed osofsky author introduced studied particular extension case obtain galois coverings enveloping algebras trivial extension algebras triangular algebras also master thesis introduced studied factorization properties extension trivial extension ring paper introduce following extension ring construction arbitrary integer let family family bilinear maps written multiplicatively particular submodules bilinear maps specified multiplication see examples section extension set denoted simply whose anderson underlying additive group multiplication given could also define product homomorphism see section details sake simplicity convenient set follows ambiguity arises extension simply called extension denoted simply general need commutative ring section give conditions maps force ring unless otherwise stated assume maps defined commutative associative ring identity thus commutative ring identity moreover naturally isomorphic subring generalized triangular matrix ring consisting every ring naturally isomorphic every particular free basis also naturally isomorphic extension set indeterminates namely field every matrices upper triangular toeplitz matrices author used terminology matrices entries commutative ring extensions rings also trivial extension ring ideal connected rees algebra associated precisely following graded subring indeterminate using lemma proposition get similar proposition following diagram extensions isomorphisms rings paper study properties ring extending results classical trivial extension rings paper organized follows section carefully define extension giving conditions maps actually commutative ring identity actually investigate situation greater generality assumed commutative end section number examples section investigate constructions extensions begin showing may considered graded ring three different grading monoids particular may considered ring ring show behaves respect polynomials corollary power series theorem extensions localization theorem theorem show extension finite direct product rings finite direct product extensions end two results inverse limits direct limits extensions theorems section present natural ring homomorphisms related extensions see proposition also study basic properties namely extend characterization prime maximal ideals classical trivial extension see theorem consequence nilradical jacobson radical determined see corollary finally extension theorems set zero divisors set units set idempotents also characterized see proposition section investigate graded aspect extensions motivation behind study classical case study trivial extensions rings lead interesting properties see shed light structure ideals trivial extensions section extend results given anderson give new ones namely among results characterize homogeneous ideals theorem investigate properties propositions devote remainder section investigate question every ideal given class ideals homogeneous see discussion proposition context various results examples established section devoted classical properties namely characterize respectively noetherian artinian manis valuation chained arithmetical generalized pir end section remark question posed concerning rings finally section study divisibility properties extensions mainly interested showing one could extend results classical trivial extension presented section context extensions general extension construction examples purpose section formally define extension commutative ring identity give interesting examples extensions however better understand construction underlying multiplication maps begin general context associative ring necessarily commutative identity also significant difference cases handle three cases separately let associative ring identity unitary case commutative always assume unless stated otherwise case trivial extension multiplication associative ring identity associative distributive laws follow ring axioms commutative write ring matrix ring representation isomorphic tri mentioned introduction note could drop assumption identity unitary get identity namely identity unitary extensions rings case addition multiplication map readily see satisfying distributive laws equivalent additive coordinate since assumed associative associative precisely equivalent call function map additive coordinate middle linear left right homogeneous note map uniquely corresponds homomorphism thus map corresponds homomorphism could equivalently define associative ring identity precisely map homomorphism identify matrix representation given introduction identified relationship tensor algebra symmetric algebra difficult associative ring define ring epimorphism tri commutative case get similar ring epimorphism commutative ring identity need commutative identity thus anderson commutative commutative ring symmetric map equivalently case associative ring identity addition assume maps equivalently corresponding homomorphism usual set setting write multiplication satisfies distributive laws maps additive coordinate necessarily associative ring identity see example case associative note associative precisely terms maps says equivalently idmk idmi idml identity map words diagram commutes idmi let call family maps satisfying viously stated associativity condition family product maps equivalently family product maps associative ring identity commutative ring identity need commutative identity every equivalently flip map defined every words diagram extensions rings commutes case family called family commutative product maps commutative equivalently family commutative product maps commutative ring identity case associative ring identity identify matrix representation given introduction identified also case associative ring define ring epimorphism tri similar result concerning symmetric algebra commutative remark let two rings every generalized triangular matrix ring naturally isomorphic trivial extension actions defined follows every see observation product two matrices generalized triangular matrix ring shows fact extended extensions consider generalized triangular matrix ring anderson family rings family modules assume every exists homomorphism denoted multiplicatively every consider finite direct product rings set need define action family product maps extension isomorphic first note every matrix every diagonal main diagonal naturally corresponds following tuple hand consider two matrices denote product using correspondence diagonal main diagonal seen every thus cases allow define left right actions follows every extensions rings cases used define product maps follows fix consider therefore endowed products extension naturally isomorphic generalized triangular matrix ring known generalized triangular matrix ring seen generalized triangular matrix namely natural ring isomorphism however extension necessarily extension consider instance extension one check easily isomorphic extension end section number examples example suppose commutative ring consider family product maps rei rej place suppose commuting indeterminates case get case interesting put rei sej rri commutative associative thus commutative also associative however commutative integral domain indebted rozas universidad spain pointed remark anderson associative commutative thus take ring reader easily check commutative associative thus commutative integral domain thus associative forces commutative case associative thus three numbers given nonzero one possible choice remaining associative take resulting ring commutative associative reader easily write conditions commutative associative example let commutative ring using multiplication ring commutative commutative following interesting special cases let commutative ring ideal quotient rees ring mentioned introduction let commutative ring ideals example extension since example could take suppose commutative ring let particular could take multiplication induced ring example could take polynomials degree example let commutative ring let usual ring product zero map considered extensions rings example let commutative ring let ideals take product given example let commutative ring ideals cyclic consider products usual products define follows adopt following notation notation unless specified otherwise denotes ring integer family bilinear maps indicated definition extension defined commutative associative ring identity indeed commutative ring identity let nonempty subset family sets every subset denoted simply constructions extensions section investigate constructions extensions first investigate graded aspect extensions convenience reader recall definition graded rings let commutative additive monoid recall ring said ring family subgroups abelian group said abelian group note subring ring module simply called graded ring graded module see instance details graded rings although deals group graded rings may considered graded ring following three different grading monoids ring case set extend definition follows rmj anderson multiplication respectively rerspectively define thus ring ring case consider least nonnegative integer mod set mba define maps zero map ring ring commutative monoid addition isomorphic case define maps zero map ring note gradings set homogeneous elements observed ring graded ring isomorphic following result presents converse implication namely shows extensions realised quotients graded rings proposition let ring product induced naturally ring isomorphic proof obvious following result presents particular case proposition proposition following two natural ring isomorphisms tri trn moreover suppose free basis graded isomorphic particular natural maps isomorphic proof obvious extensions rings next result shows extension graded ring graded modules natural grading extension theorem theorem let commutative additive monoid assume every ring proof similar proof theorem case either polynomial ring laurent polynomial ring get following result first assertion extension corollary corollary following statements true set indeterminates set indeterminates also classical case get related graded power series case generalization corollary first recall given set analytic indeterminates consider three types power series rings see details generalized power series rings homogeneous degree possibly inf inite sum monomials degree one monomial orm set generally given partially ordered additive monoid generalized power supp series ring consists formal sums artinian narrow infinite family incomparable elements addition multiplication carried usual way set reverse lexicographic order note similar manner define three types power series module routine proof following theorem left reader anderson theorem let set analytic indeterminates let partially ordered additive monoid give extension theorem following result investigates localization extension need following technical lemma lemma every every every consequently element denote element lemma theorem let multiplicatively closed subset family rmodules submodule every set multiplicatively closed subset ring isomorphism mis proof trivial show multiplicatively closed subset order show desired isomorphism need make done proof theorem following observation let using notation lemma sei sei map desired isomorphism extensions rings simple important particular case theorem get following result extends theorem corollary theorem show prime ideal prime ideal fact used next result show localization extension prime ideal isomorphic extension follows use denote total quotient ring ring proposition prove set regular elements thus corollary following assertions true let prime ideal ring isomorphism mip ring isomorphism indeterminate ring isomorphism proof proofs similar corresponding ones classical case next result generalizes theorem shows extension finite direct product rings finite direct product extensions reader convenience recall known facts structure modules finite direct product finite direct product rings set rings let submodule namely every element written form every place note also ismorphic via following anderson consider family commutative product maps define following maps easily checked every family commutative product maps extension furthermore notation mind ready give desired result theorem let finite direct product rings proof easily checked map isomorphism end section two results investigate inverse limit direct limit system extensions namely show conditions inverse limit direct limit system extensions isomorphic extension inverse limit case generalization theorem let directed set inverse system abelian groups know inverse limit isomorphic following subset direct product next result mean exactly set theorem let directed set integer consider family inverse systems satisfy following conditions every ring extensions rings every every every extension family commutative product maps satisfy every limmn extension following family commutative product maps limmi limmj moreover natural ring ismorphism lim limmn proof result follows using standard argument let directed set direct system abelian groups know direct limit lim isomorphic generated elements natural inclusion map since directed every element form theorem let directed set integer consider family direct systems satisfy following conditions every ring every every every extension family commutative product maps satisfy every anderson limmn extension following family commutative product maps limmi limmj moreover natural ring ismorphism lim limmn proof result follows using standard argument basic algebraic properties section give basic properties extensions giving first result make following observations situations subfamily trivial observation integer every natural ring isomorphism represented however integer general represented extension indeed example satisfies makes sense since subset every represented trivial extension biggest odd integer namely natural ring isomorphism every natural ring isomorphism extensions rings biggest even integer general every cyclic submonoid generated element natural ring isomorphism msg biggest integer observed one would discuss according whether subfamily trivial various situations may occur thus sake simplicity make following convention convention unless explicitly stated otherwise consider extension given implicitly suppose every used sequel without explicit mention note also nature maps affect structure extension example case example extension ideals let start following result presents relations easily established extensions proposition following assertions true let submonoid consider family following natural ring extensions particular every following natural ring extensions extension denoted every ideal every via action moreover structure ideal structure every particular structure ideal one anderson every following natural ring isomorphisms obtained natural ring homomorphism give another example assertion one show submonoid following natural ring extensions remark seen case ideal structure structure actually nagata used reduce proofs results ideal case however structure need ideal structure instance consider extension maps induced multiplication ideal generated however according proposition notion extensions ideals ring homomorphisms natural way construct examples ideals context use ring homomorphism indicated proposition give examples proposition ideal following assertions ideal imn extension ring homomorphism following natural ring isomorphism imn multiplications follows ideal imn finitely generated finitely generated proof proof straightforward using clear generated elements set generated conversely generated elements set imn generated extensions rings determine radical prime maximal ideals classical case show ideals particular cases homogenous ones characterized next section however give particular cases simplicity reflected using following lemma fact contain nilpotent ideal index lemma every ideal contains form ideal case following natural ring isomorphism proof let ideal contains consider ideal surjective ring homomorphism used proposition fact deduce finally using fact get desired isomorphism following result extension theorem theorem radical ideals form radical ideal particular maximal prime ideals form resp maximal prime ideal proof using lemma sufficient note every radical ideal contains since theorem allows easily determine jacobson radical nilradical corollary jacobson radical nilradical nil nil krull dimension equal end section extension theorems determines respectively set zero divisors set units set idempotents worth noting trivial extensions used construct examples rings zero divisors satisfies certain properties mentioned introduction particular extensions used settle questions recently extension used context graphs give appropriate example proposition following assertions true set zero divisors anderson hence set regular elements set units set idempotents proof proofs similar corresponding ones classical case completeness give proof first assertion let every hence suppose exists nonzero element hence exists nonzero element hence since union prime ideals nil contained prime ideal using fact nil conclude gives first inclusion conversely let smn get otherwise pass continue arrive gives desired inclusion homogeneous ideals extensions study classical trivial extension graded ring established interesting properties see instance section namely studying homogeneous ideals trivial extension shed light structure ideals naturally one would like extend study context extensions section extend study context extensions graded ring indicated section every every note could also consider ring ring mentioned section convenient recall following definitions let commutative additive monoid ring let every extensions rings elements said homogeneous degree submodule said homogeneous one following equivalent assertions true generated homogeneous elements finite subset homogeneous degree every particular ideal homogeneous note ideal satisfies imi evey next result extends theorem namely determines structure homogeneous ideals extensions follows use ring homomorphism used proposition following homomorphism theorem following assertions true let ideal let family every homogeneous ideal imi thus ideal every natural ring isomorphism multiplications follows particular let ideal consider every anderson ideal submodule every kmi every thus homogeneous ideal ideal homogeneous proof ideal imn thus imi every conversely suppose imi every map surjective homomorphism kerf ideal particular three statements easily checked following result presents properties homogeneous ideals extension theorem theorem particular determine extension theorem form homogeneous principal ideals fact characterization homogeneous principal ideals plays key role studying homogeneous ideals due easily checked fact ideal graded ring homogeneous every principal ideal generated element homogeneous proposition following assertions true let two homogeneous ideals following homogeneous ideals inn extensions rings every principal ideal homogeneous rmn amn ideal particular homogeneous ideal proof proof first three statements similar corresponding one theorem last statement easily follows fact residual two homogeneous ideals homogeneous apply assertion theorem proof similar one theorem known fact case even homogeneous ideal finitely generated necessarily finitely generated consider principal ideal example following result presents context particular cases obtained using standard arguments proposition following assertions true ideal finitely generated finitely generated homogeneous ideal finitely generated finitely generated ideal converse implication true finitely generated every previous section note every radical hence prime ideal homogeneous however ideals classical trivial extensions general homogeneous see natural questions arise question every ideal given class ideals homogeneous question given ring family class homogeneous ideals clear questions depend structure instance ring maximal proper homogeneous ideal anderson either form proper ideal proper homogeneous principal ideal either form principal ideal instance principal ideal generated element nonzero homogeneous question investigated case class regular ideals theorem also condition integral domain characterization trivial extension rings every ideal homogeneous given see theorem corollary aim remainder section extend study ntrivial extensions worth noting classical case ideals homogeneous shows condition ideals homogeneous implies ideals homogeneous context show situations occur let begin class ideals given subset regular elements recall ring said every implies rings introduced studied bouvier series papers see references also investigated notion ring used homogeneous ideals classical trivial extensions studied example integral domain every ideal homogeneous see theorems first consider subsets regular elements theorem let nonempty subset let class ideals following assertions equivalent every ideal homogeneous every principal ideal homogeneous every smi every principal ideal form principal ideal every ideal form ideal proof obvious let need prove smi consider element since homogeneous extensions rings since regular desired let principal ideal implies using inclusion get inductively get every thus therefore lemma proposition form consider ideal element therefore using lemma get desired result obvious example consider trivial extension ring fractions respect multiplicatively closed subset principal ideal homogeneous deny must thus implies absurd following result extension theorem recall ideal said regular contains regular element proposition ideal regular contains element corollary let following assertions equivalent every regular ideal homogeneous every principal regular ideal homogeneous every smi equivalently mis every principal ideal form principal ideal every regular ideal form ideal consequently root closed particular integrally closed every regular ideal form given proof proof similar one theorem anderson compare following result corollary corollary assume integral domain following assertions equivalent every ideal homogeneous every principal ideal homogeneous every divisible every principal ideal form nonzero principal ideal every ideal form nonzero ideal every ideal comparable proof equivalence simple consequence lemma proof theorem shows another situation considered given following result use annr denote annihilator theorem let class ideals nonempty subset every annr annr following assertions equivalent every ideal homogeneous every principal ideal homogeneous every smi every principal ideal form principal ideal every ideal form ideal proof need prove implication let consider element since homogeneous hypothesis therefore xmi desired extensions rings example ring satisfies condition previous result consider ring idempotent set every thus since emi every every ideal homogeneous unlike classical case fact every divisible necessarily imply every ideal homogeneous consider extension field principal ideal homogeneous indeed homogeneous must thus implies absurd example naturally leads investigate every ideal homogeneous context notion module used recall called every implies example integral domain every module see also studying question every ideal homogeneous several different cases occur use following lemma lemma let ideal following assertions true ideal homogeneous form homogeneous form proof straightforward theorem assume given let class ideals every following assertions equivalent every ideal homogeneous every principal ideal homogeneous every every every principal ideal form nonzero cyclic submodule anderson every ideal form nonzero submodule every ideal contains proof implication proved similarly implication theorem implication simple consequence lemma implication needs proof let principal ideal homogeneous implies exists rmj rmk since invertible desired examples rings satisfy conditions previous result consider following two extensions ring fractions respect multiplicatively closed subset following particular cases interest corollary assume let class ideals every following assertions equivalent every ideal homogeneous every every ideal comparable proof particular case corresponding one theorem let ideal theorem shows contains otherwise means contains let nonzero ideal theorem shows homogeneous case consequence assertion lemma get following particular case corollary corollary assume let class ideals following assertions equivalent extensions rings every ideal homogeneous every every ideal comparable theorem additional conditions equivalent study case leads introduce following notion order avoid trivial situations definition assume product said family multiplications elements mij product mij least one mik zero ambiguity arises simply called corollary assume every let class ideals following assertions equivalent every ideal homogeneous every principal ideal homogeneous every every every every nonzero elements every every nonzero element every principal ideal form nonzero cyclic submodule every ideal form nonzero submodule every ideal contains proof equivalences easily proved following result shows fact conditions corollary necessary sufficient show every ideal homogeneous note corollary presents case thus following result may assume corollary assume every following assertions equivalent anderson every ideal homogeneous every every ideal homogeneous proof need prove let consider let rmj corollary implies since every therefore invertible since prove every ideal homogeneous suffices prove every every every theorem case trivial thus fix consider prove corollary corollary also corollary desired finally give case characterize rings every ideal homogeneous note ring ami every every every every nonzero element integral domain must every corollary suppose integral domain assume torsionfree every every following assertions equivalent every ideal homogeneous following two conditions satisfied every divisible every every proof simply use corollaries theorem easy show two extensions satisfy conditions result every ideal rings homogeneous end section following particular case extensions rings corollary suppose consider extension field space suppose every following assertions equivalent every ideal homogeneous every every every particular case consider field extension every ideal homogeneous namely every proper ideal form properties section determine certain ring properties noetherian artinian manis valuation chained arithmetical generalized pir end section remark question posed concerning rings begin characterizing extensions noetherian artinian following result extends theorem theorem ring noetherian artinian noetherian artinian every finitely generated proof similar proof theorem following result extension theorem corollary investigates integral closure total quotient ring theorem let integral closure mns integral closure particular integrally closed ring mns integral closure mns integrally closed integrally closed proof statements proved similarly corresponding ones theorem corollary anderson worth noting classical case integrally closed without integrally closed see example given corollary similar classical case theorem consequence theorem corollary give following result characterizes manis valuation first recall two notions let subring ring let prime ideal called valuation pair valuation ring surjective valuation min totally ordered abelian group equivalent exists valuation ring called manis valuation ring also called ring every finitely generated regular ideal invertible equivalent every overring integrally closed see details corollary let manis valuation ring valuation ring mis every ring every finitely generated ideal invertible mis every extension theorem characterize chained ring recall ring said chained set ideals totally ordered inclusion exception convention following results lemma theorem corollary lemma theorem module family associated extension zero proof desired result uses following lemma gives another characterization particular extension property every ideal homogeneous lemma assume maximal ideal suppose also least one modules family nonzero every ideal homogeneous following three conditions satisfied integral domain every divisible every emi every case ideal one forms ideal nonzero submodule extensions rings proof clearly first assertion simple consequence second one need prove second third assertions let consider element maximal ideal invertible trivially get result next assume hypothesis ideal homogeneous thus invertible thus desired let consider hypothesis homogeneous implies get impossible since either invertible suppose exists hence using fact equality becomes previous case impossible therefore gives desired result need prove every principal ideal homogeneous distinguish two cases follow argument similar theorem theorem assume least one modules family nonzero ring chained following conditions satisfied valuation domain every divisible every emi every every set cyclic submodules totally ordered inclusion anderson proof first prove chained ring consider two ideals two ideals comparable desired similar argument used prove last assertion prove every ideal homogeneous lemma get assertions consider nonzero ideal necessarily lemma homogeneous let smallest integer first assertion lemma homogeneous necessarily thus second assertion lemma homogeneous desired using lemma deduce two ideals forms submodules respectively obviously comparable using first last assertion show comparable desired using theorem corollary get extension theorem characterizes arithmetical recall ring arithmetical chained prime maximal ideal also recall ring called arithmetical prime maximal ideal set submodules totally ordered inclusion finally recall support supp ring set prime ideals corollary ring arithmetical following conditions satisfied arithmetical every arithmetical every valuation domain every every supp mip divisible every supp supp supp emip mjp every recall ring called generalized every proper ideal proper principal ideal product prime ideals integral domain called clearly generalized nothing dedekind domain well known example see sections generalized principal ideal ring pir finite direct product extensions rings following types rings dedekind domains pids fields special principal ideal rings spirs fields next results extend lemma theorem characterize generalized pir lemma generalized pir generalized pir hence either dedekind domain pid field spir field let domain spir field field conversely generalized pir generalized pir proof using theorem proof similar lemma theorem generalized pir generalized pir cyclic annihilator pis pis idempotent maximal ideals ann proof similar proof theorem end section remark question posed recall ring called char thus boolean rings rings shown theorem rings represented trivial extensions namely proved nil theorem based result following natural question posed see page whether theorem extended rings one ask whether extension suitable construction solve question using theorem one show amalgamated algebras along ideal introduced resolve partially question recall given ring homomorphism ideal amalgamation along respect following subring ring whose underlying group note multiplication given via see details theorems nil nil anderson canonical injection actually extension seen amalgamation along respect canonical injection leads pose following question every ring extension divisibility properties factorization commutative rings zero divisors first investigated series papers bouvier fletcher billis see references focus unicity property papers marked start systematic study factorization commutative rings zero divisors since theory attracted interest number authors study divisibility properties classical trivial extension lead interesting examples answering several questions see section section interested extending part study context extensions first recall following definitions let commutative ring two elements said associates written strong associates written strong associates written either implies taking gives notions associates say strongly associate every strongly associate also say strongly associate finally recall said say begin extension theorem proposition let ring extension every particular suppose nilpotent ideal satisfies either graded ring proof let consider since let two elements assume implies since nilpotent invertible desired next assume case extensions rings hence since finally case graded ring may set nim implies similarly proposition let graded ring strongly associate strongly associate ring strongly associate every suppose exists every assume ring strongly associate strongly associate proof let rxi ryi hence uyi desired let two associate elements nonzero particular hence two cases occur case since desired case result follows since strongly associate give extension theorem context extensions corollary following assertions true strongly associate strongly associate suppose strongly associate strongly associate investigate extension theorem convenient recall following definitions let commutative ring nonunit said irreducible atom strongly irreducible strongly irreducible implies said maximal element set proper principal ideals note nonzero nonunit strongly irreducible strongly irreducible irreducible none implications reversed case say strongly strongly anderson implies note maximal cyclic strongly strongly ann strongly ann following results homogeneous element denoted following result extends theorem proposition let proof assertion proved similarly corresponding classical one worth noting analogue assertion theorem hold context extensions indeed consider extension easy show superprimitive however strongly irreducible since moreover even assume integral domain still desired analogue take superprimitive however strongly irreducible since last example also shows assertion theorem hold context extensions namely irreducible indeed neither hmi extend theorem need introduce following definitions definition assume multiplication family trivial let element said indecomposable relative family multiplications every ambiguity arise elements simply called indecomposables example every element indecomposable however every element submodule decomposable since definition let said integral relative family multiplications every implies ambiguity arise simply called integral extensions rings example integral integral since instance proposition assume let irreducible strongly irreducible strongly irreducible primitive strongly primitive strongly primitive conversely three cases occur case reverse implication holds integral domain every case reverse implication holds integral domain every indecomposable case reverse implication holds integral domain every integral every indecomposable proof prove primitive irreducible case two cases proved similarly suppose irreducible let ani implies rmi rni desired let first show case impossible cases easy left reader assume suppose following equalities recursive argument equalities shows indeed clear true suppose true given equality becomes since integral get desired result thus get absurd since indecomposable may assume since recursively get since primitive exists remains show implies desired using get since torsionfree set similarly using equality equalities get finally recursive argument gives desired result anderson following result extends theorem proposition suppose nontrivial idempotent every irreducible proof assertion proved similarly corresponding classical one interested factorization properties recall ring called atomic every nonzero nonunit product irreducible elements atoms note domain case ascending chain condition principal ideals accp implies atomic begin extension theorem characterizes trivial extension ring satisfies accp need following lemma lemma let consider two elements implication hai hbi hai hbi true either proof since hai hbi shows hai hbi cases theorem assume suppose every satisfies accp satisfies accp every satisfies acc cyclic submodules proof proof direct implication easy let prove converse suppose admits strictly ascending chain principal ideals exists every lemma get following strictly ascending chain principal ideals absurd since satisfies accp suppose every exists also lemma obtain following strictly ascending chain cyclic submodules absurd since satisfies acc cyclic submodules continue way case may suppose every every therefore lemma get desired result extensions rings investigate atomic namely give extension theorem recall said satisfy mcc every cyclic submodule contained maximal necessarily proper cyclic submodule theorem assume suppose every atomic satisfies accp satisfies acc cyclic submodules every satisfies mcc proof proof slightly technical one theorem need break proof following steps step number prove every nonunit element product irreducibles use inductive argument first steps step suppose nonunit element factored irreducibles exist neither associate since clearly must reducible say also neither associate say reducible continue obtain strictly ascending chain using lemma get strictly ascending chain principal ideals absurd since satisfies accp step suppose nonunit element product irreducibles neither associate every necessarily hence preceding steps product irreducibles hypothesis reducible product irreducibles reducible thus symmetry may assume reducible product irreducibles necessarily repeat last argument obtain strictly ascending chain principal ideals anderson every lemma get strictly ascending chain cyclic submodules absurd hypothesis desired step remains prove every element form product irreducibles since satisfies mcc rmn maximal cyclic submodule shows product irreducibles step maximal shows either irreducible preceding steps products irreducibles hence concludes proof ring said bounded factorial ring bfr nonzero nonunit natural number factorization nonunit domains say bfd instead bfr recall said nonzero exists natural number nonunit next theorem generalization theorem investigates bfr based following lemma lemma product elements form form product zero theorem assume integral domain every bfr bfd every proof clear let nonzero nonunit element suppose factorization nonunits implies otherwise may assume since integral domain lemma may assume every every let since every therefore since extensions rings investigate notion introduced fletcher developed axtell let ring consider nonunit factorization mean nonunit recall rsa factorization every every denote call irrelevant relevant factors next result investigates extension first recall following definitions ring called finite factorization ring ffr factorization ring every nonzero nonunit finite number factorizations order associates associates relevant factors ring called weak finite factorization ring wffr finite factorization ring every nonzero nonunit finite number nonassociate divisors nonassociate relevant factors ffr wffr converse holds domain case wffr ffr however theorem study notions classical trivial extensions lead consider following notion see let nonzero element say reduced submodule factorization every cancelling reordering case module said module every nonzero element exist finitely many reduced submodule factorizations order associates well associates context introduce following definition definition assume consider let dis say dis submodule factorization every dij cancelling reordering case dis dit ambiguity arise submodule factorization simply called reduced submodule factorization said module simply module every nonzero element exist finitely many reduced submodule factorizations dis order associates dij clear notion module defined axtell one anderson based proof theorem asserted theorem every nonzero nonunit finitely many distinct principal ideals however careful reading proof shows case ideals also treated one confirm validity assertion reduced rings however context extensions seems complicated nevertheless certain conditions next investegate lemma assume integral every nonzero nonunit following assertions true every following assertions equivalent dmi dmi following assertions equivalent dmi dmi every dmn dmi every every proof implications nothing prove let dmi therefore since integral similar previous proof implication nothing prove first prove every therefore since integral consider every dmj shows dmj proved since since integral absurd since finally assertions conclude every extensions rings theorem assume equivalently following conditions satisfied ffr module every moreover integral domain integral every nonzero nonunit finitely many distinct principal ideals every every finitely many distinct principal ideals conversely integral domain integral assertions imply proof proof converse part similar corresponding one theorem proof similar given theorem suppose contradiction exists nonzero nonunit family distinct principal ideals form infinite indexing set prove impossible showing every exists recursive argument shows fact every equation admits solution implies existence desired note lemma every first consider element every dbn dbn possible corresponding dbn since exists drbn shows equation admits solution consider every drbk drbk therefore drbk since integral let suppose contradiction exists family distinct principal ideals form infinite indexing set let necessarily possible corresponding since exists anderson equivalently absurd ring called factorization ring nonzero nonunit natural number factorization said module every exists natural number cancellation reordering question classical trivial extension still open however answer question integral domain theorem bfd two general results direct implication established theorem lemma extend results context need introduce following definition definition assume consider said module simply module every nonzero element exists natural number dit dit mit dij cancellation reordering dij dis theorem module every moreover bfr conversely assume integral domain bfd every module proof similar classical case ring called every nonzero nonunit element relevant factors irreducibles question classical trivial extension still unsolved theorem axtell gave answer question integral domain accp atomic theorem shown condition ring integral domain could replaced ring following result gives extension theorem context extensions theorem assume suppose every satisfies accp satisfies acc cyclic submodules every atomic proof clear suppose atomic proof theorem exists product irreducibles since extensions rings admits form irreducibles since product irreducibles proof theorem necessarily form form impossible since references anderson factorization commutative rings zero divisors iii rocky mountain math anderson generalizations boolean rings rings von neumann regular rings comment math univ paul anderson axtell forman stickles associates unit multiples rocky mountain math anderson markanda unique factorization rings zero divisors houston math anderson markanda unique factorization rings zero divisors corrigendum houston math anderson pascual regular ideals commutative rings sublattices regular ideals rings algebra anderson factorization commutative rings zero divisors rocky mountain math anderson factorization commutative rings zero divisors factorization integral domains lecture notes pure appl marcel dekker new york anderson winders idealization module commut algebra ara nicholson yousif look glasgow math axtell commutative rings zero divisors comm algebra axtell forman roersma stickles properties int commut rings anderson barucci anna strazzanti family quotients rees algebra comm algebra bennis mikram taraza extended zero divisor graph commutative rings turk math billis unique factorization integers modulo amer math monthly birkenmeier heatherly kim park triangular matrix representations algebra birkenmeier park rizvi extensions rings modules springer media new york bouvier type commutatifs factorisation unique acad sci paris bouvier type acad sci paris bouvier factorisation dans les fractions acad sci paris bouvier factorisation dans les acad sci paris bouvier anneaux anneaux atomiques acad sci paris bouvier sur les anneaux fractions des anneaux atomiques bull sci math bouvier sur les anneaux fractions des anneaux atomiques bull sci math bouvier anneaux gauss acad sci paris bouvier remarques sur facatorisation dans les anneaux commutatifs pub math lyon bouvier nouveaux sur les anneaux acad sci paris bouvier anneaux acad sci paris extensions rings bouvier nouveaux sur les anneaux acad sci paris bouvier anneaux rev roumaine math pures appl bouvier structure des anneaux factorisation unique pub math lyon bouvier sur les anneaux principaux acta math sci hung camillo herzog nielsen injective hulls compatible multiplication algebra anna finocchiaro fontana amalgamated algebras along ideal commutative algebra applications proceedings fifth international fez conference commutative algebra applications fez morocco gruyter publisher berlin fletcher unique factorization rings proc camb phil soc fletcher structure unique factorization rings proc camb phil soc fletcher euclidean rings london math soc huckaba commutative rings zero divisors monographs textbooks pure applied mathematics marcel dekker new york mcqueen factorization rings toeplitz matrices master thesis university central missouri nagata local rings interscience publishers new nastasescu van oystaeyen methods graded rings lecture notes math berlin northcott lessons rings modules multiplicities cambridge univ press cambridge pogorzaly generalization trivial extension algebras pure appl algebra ribenboim rings generalized power series units algebra
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fft algorithm binary extension finite fields application codes aug lin member ieee tareq member ieee yunghsiang han fellow ieee specifically since practical implementations codes typically binary extension finite fields complexity codes fields received attentions others new polynomial basis binary extension fields proposed fast fourier transform fft fields computed complexity order number points evaluated fft work reformulate fft algorithm easier understood extended develop frequencydomain decoding algorithms systematic reedsolomon codes power two first basis syndrome polynomials reformulated decoding procedure new transforms applied decoding procedure fast extended euclidean algorithm developed determine error locator polynomial computational complexity proposed decoding algorithm improving upon best currently available decoding complexity reaching best known complexity bound established justesen however justesen approach codes specific fields apply ffts revealed computer simulations proposed decoding algorithm times faster conventional one code conventional decoding algorithm quadratic complexities fast approaches based ffts fast polynomial arithmetic techniques however structures ffts finite fields vary sizes fields smooth number meaning factorized many small primes fft field additions field multiplications applied conventional case involves choosing fermat primes based ffts justesen gave approach decoding code another approach solve key equations bch codes proposed pan reduces factor characteristic field large enough however algorithm improvement codes binary extension fields smooth ffts inapplicable case ffts arbitrary fields applied requires field operations gao presented decoding algorithm arbitrary fields utilizing fast polynomial multiplications codes additive fft requires operations applied reduce leading constant authors knowledge additive fft fastest algorithm far ntroduction codes class block errorcorrecting codes invented reed solomon code constructed extended version called extended reedsolomon codes admits codeword length systematic version code appends parity symbols message symbols forming codeword length codes maximum distance separable mds codes correct erroneous symbols nowadays codes numerous important applications including barcodes codes storage devices discs digital television dvb atsc data transmission technologies dsl wimax codes also used design forward error correction codes regenerating codes local reconstruction codes wide range applications codes raises important issue concerning computational complexity codes typically constructed binary extension fields consider case paper clearly one wants remove extra factor algorithms binary extension fields ffts required recently lin showed new way solve aforementioned fft problem paper defined new polynomial basis based subspace polynomials polynomial degree less new basis multipoint evaluations made field operations based multipoint evaluation algorithm decoding algorithms codes proposed achieve however errorcorrection decoding algorithm based new basis yet provided work supported part cas pioneer hundred talents program national science council nsc taiwan grants nsc nsc lin school information science technology university science technology china ustc hefei china electrical engineering department king abdullah university science technology kaust kingdom saudi arabia sjlin tareq electrial engineering department king abdullah university science technology kaust thuwal makkah province kingdom saudi arabia han department electrical engineering national taiwan university science technology taipei taiwan yshan paper develops error correction decoding algorithm codes power practice codes usually rates complexity proposed algorithm given holding constant code rate yields complexity better best existing complexity achieved gao algorithm based polynomial basis embed new basis decoding algorithm reformulate decoding formulas arithmetics performed new basis key equation solved euclidean algorithm thus fast polynomial divisions well euclidean algorithm new basis proposed finally combine algorithms resulting fast errorcorrection decoding algorithm major contributions paper summarized follows alternative description algorithms new polynomial basis presented fast polynomial division new basis derived fast algorithm new basis presented encoding algorithm presented power two decoding algorithm based new basis demonstrated decoding algorithm presented power two notably gave encoding algorithms codes complexity power two encoding algorithm suitable coding rate however proposed encoding algorithm work suitable rest paper organized follows section reviews definitions polynomial basis multipoint evaluation algorithm provided sec iii section provides alternative polynomial basis constructed using monic polynomials polynomial operations used codes explicated section presents fast extended euclidean algorithm based method section section vii introduce algorithms encoding decoding codes section viii presents simulations draws conclusions basis space form strictly ascending chain subspaces given let denote elements element defined binary representation implies note additive identity filed work used interchangeably confusion subspace polynomial defined clear see deg example properties given theorem polynomial implies formal derivative constant recursive form subspace polynomials given let denote basis defined section reviews subspace polynomials polynomial basis defined subspace polynomial let denote extension finite field dimension let denote basis linearly independent space defined polynomial basis olynomial basis span binary representation notice example seen deg thus basis represent elements many chosen algorithm transform basis input binary logarithm size output evaluations return end algorithm inverse transform basis input binary logarithm size output coefficients return end call call end return end call call return polynomial degree basis represented divided two subsets algorithm relied following lemma throughout paper used indicate vector coefficients due fact deg new basis possesses following properties corollary given polynomial respectively expressed monmial basis lemma given polynomial basis based lemma algorithm compute described substituting obtain following properties hold determined vice versa iii ultipoint evaluations polynomial set let notation denote set evaluation values gave recursive algorithm calculate converts furthermore substituting obtain section describe algorithm another viewpoint helps develop algorithm codes set evaluation points divided two individual subsets converts set evaluation points expressed coset adding accordingly set polynomial evaluations comparing degrees polynomials reduced number terms reduced complexity obtaining polynomials discussed coefficient takes addition multiplication except without arithmetic operations however consider exception reduction exceptions limited coefficients takes total additions multiplications obtain calculating coefficient takes addition takes total additions obtain coefficients procedure applied recursively set size set one strategy additive complexity multiplicative complexity respectively written transform result algorithm depicts details recursive approach denoted inverse fft obtained backtracking fft given opposite inverse transform get coefficients objective find coefficients reformulate compute coefficients coefficients obtained applying inverse transform recursively details shown algorithm note denotes inverse transform algorithms use notations one follow easily clear algorithms number arithmetic operations figure showed example proposed algorithm inversion pits input polynomial defined output given inverse transform fig data flow diagram proposed transform inversion olynomial basis monic polynomials objective multiplications align results desired polynomial extracted properly let denote quotient dividing basis deg precisely polynomial degree operations section define alternative version polynomial basis algorithms perform multiplications formal derivatives divisions new basis operations used coding algorithms alternative basis defined ifftx quotient dividing general given dividend divisor division determine quotient remainder deg deg without loss generality consider case deg deg inversion denoted ifftx based algorithm transforms defined fftx basis conversion multipoint evaluation algorithm also applied complexity unchanged simplify notations rest paper polynomials represented evaluations denoted fftx given implies monic polynomial basis conversion requires proposed algorithm firstly finds quotient remainder calculated following focus algorithm determine let deg operation pairwise multiplication two vectors operation pairwise division since multiplication formal derivative similar given summarize appendix completeness next present algorithm polynomial division essential decoding codes dlg deg begin multiplied obtain polynomial division simplify notations let subsection proposed polynomial division basis proposed algorithm based newton iteration approach used fast division algorithms standard basis fft exists however since basis different standard basis moderate modifications required compared conventional fast division proposed approach two major differences first conventional fast division shall reverse coefficients divisor upon performing newton iteration however basis polynomial reversion applied thus proposed algorithm reverse polynomials operations performed polynomials without reversions second proposed algorithm includes specific multiplications required conventional approach next present method determine assume exists polynomial sda deg deg dlg deg algorithm polynomial divisions input dividend divisor deg deg output quotient remainder algorithm find addressed section determining first present two lemmas whose proofs given appendix lemma lemma degree thus compute deg deg sda deg deg obtaining multiplied obtain defined find holds compute compute return updated polynomial degree calculated initial polynomial sda sda let lemma side degree rewritten deg polynomial coefficients starts sda lemma degree side thus quotient starting degree hence quotient obtained deg residual clearly deg defined verified deg deg holds validity supported follows proofs given appendix deg algorithm shows steps division algorithm complexity analyzed step deg deg deg deg deg deg complexity step show suffice section step shows complexity step degrees polynomials less complexity summary algorithm complexity deg deg lemma possesses following equality deg following reformulation contains polynomial multiplications used determine complexity calculating lemma rewritten determining given pdb given bdb subsection presents method find notice deg proposed method seen modified version division newton iterations method iteratively computes coefficients highest degree lowest degree algorithm depicts steps algorithm repeats performing obtain desired output complexity iteration lines calculates deg algorithm algorithm input hgcd basis deg deg deg output two matrices given deg return algorithm computation input polynomial output polynomial holds deg deg let compute compute equivalently end return deg deg multiplications requires lemma showed computation reduced without polynomial multiplications thus iteration takes operations complexity loop line takes end hgcd compute deg zml return zml end divided get xtended uclidean lgorithm based alf pproach section introduces extended euclidean algorithm used decoding codes given two polynomials deg deg deg deg divided three polynomials denoted rmlh rmh euclidean algorithm procedure recursively divide get compute deg deg procedure stops greatest common divisor gcd extension version namely extended euclidean algorithm calculates pair polynomials iteration rmm rmh step extended euclidean algorithm expressed matrix form next step shown algorithm calculates temporal result extended euclidean algorithm step deg hgcd rmm return rmll polynomials monomial basis exist fast approaches operations denotes complexity multiplying two polynomials degrees see algorithm figure idea comes observation quotient determined upper degree part lower degree part necessary fortunately observation also applicable basis observation partition inputs several portions procedure applied section present algorithm basis approach performed solve error locator polynomial see decoding procedure codes polynomial division requires using fast division approach sec line line polynomial additions polynomials multiplications polynomials degrees less complexity results given appendix summary overall complexity portions higher degrees algorithms monomial basis simple make partitions basis choose partition points degrees precisely divided three polynomials alh respectively representation given alh eed olomon encoding algorithm similarly partitioned manner blh section introduces encoding algorithm codes power two exist two viewpoints constructions codes termed polynomial evaluation approach generator polynomial approach polynomial evaluation approach message interpreted polynomial degree less codeword defined evaluations distinct points basis thus passume vector coefficients denoted algorithm depicts proposed algorithm hgcd deg deg deg algorithm outputs two matrices deg deg high degree part codeword computed via algorithm deg deg deg deg deg however requires operations generated codeword systematic following another formula complexity given generated codeword systematic inversion given deg deg note high degree part see begin divided number proving validity algorithm give following lemmas whose proofs given appendix lemma algorithm always outputs given satisfy elements defined lemma recursive calls hgcd meet requirements deg deg deg proved possess equality given following lemma whose proof given appendix lemma algorithm always outputs given satisfies lemma following equality hold lemma algorithm always outputs given satisfy addition vectors lemmas plays core transform proposed algorithm assume includes parity symbols others message symbols parity computed via theorem algorithm valid algorithm always outputs given satisfy four conditions determine computational complexity follows algorithm complexity denoted polynomial degrees step step algorithm shall call routine twice takes line algorithm requires fft times ifft hence complexity encoding algorithm degree term less except last term thus quotient dividing would based results new key formula vii eed olomon decoding algorithm deg key equation find error locator polynomial find extended euclidean algorithm applied extended euclidean algorithm stops remainder degree less obtaining next step find locations errors defined set roots obtaining final step calculate error values formal derivative section shows decoding algorithm codes codeword generated section proposed algorithm follows decoding process let denote received vector error pattern hence erroneous symbol suppose contains symbols let let denote polynomial degree less clear see formula leads due implies mod ifftx ifftx deg given key equation find applying euclidean algorithm however though similar key equation syndrome decoding syndrome polynomial obtain syndrome decoding new key formula quotients dividing case divided two parts notice uses compute error values rather used forney formula summary decoding algorithm consists four steps calculate syndrome polynomial determine polynomial extended euclidean algorithm find error locations calculate error values via details step described first step high degree part applying ifft received codeword however since high degree part required follow idea encoding formula particular received codeword divided several individual parts elements syndrome polynomial calculated mod substituting error value given denote set corresponding locations errors polynomial defined second step fast euclidean algorithm algorithm applied upon performing euclidean algorithm step dividing resulting call algorithm inputs obtain hgcd denotes residual notably error occurs degree less hence thus take syndrome polynomial polynomial recursively decomposed obtain thus error locator polynomial given euclidean algorithm also given paper combining algorithms fast decoding algorithm proposed achieve complexity letting constant complexity written improves upon best currently available decoding complexity although justesen given algorithm complexity include field recognized important case real applications following address potential future works remove constraint power two algorithms increase values selected generalize algorithm handle errors erasures reduce leading constant fft approach make proposed algorithm competitive short codes third step roots searched via ffts transform fftx evaluate result vector contains zeros roots corresponding points performed search roots notably deg larger number found roots decoding procedure shall terminated situation occurs number errors exceeds final step compute fftx fftx computing given appendix error values calculated via determine computational complexity first step requires times ifft complexity second step takes operations third step requires times fft thus complexity final step requires formal derivative polynomial degree times fft thus complexity summary proposed decoding algorithm requires eferences reed solomon polynomial codes certain finite fields journal society industrial applied mathematics vol wolf adding two information symbols certain nonbinary bch codes applications bell system technical journal vol sept rashmi shah kumar optimal exactregenerating codes distributed storage msr mbr points via construction ieee trans inf theory vol aug lin chung han unified form codes via frameworks ieee trans inf theory vol feb huang simitci ogus calder gopalan yekhanin erasure coding windows azure storage presented part usenix annual technical conference usenix atc boston usenix chen yan complexity analysis reedsolomon decoding without using syndromes eurasip wirel commun vol truong chen wang cheng fast transform decoding errors erasures reedsolomon codes ieee viii oncluding remarks simulations implemented algorithm compiled gcc compiler intel xeon windows platform codes program took second produce codeword tested codeword errors decoding takes seconds comparison also ran standard decoding algorithm took seconds decode codeword thus proposed decoding around times faster traditional approach parameter configurations described simulations proposed algorithm suitable long codes paper developed fast decoding algorithms systematic codes fields proposed algorithms formed new basis reformulated formulas syndromebased decoding algorithm ffts new basis applied fast polynomial division algorithm proposed made modifications newton iteration applied new basis fast trans veh vol feb justesen complexity decoding reedsolomon codes corresp ieee trans inf theory vol mar gao new algorithm decoding codes communications information network security kluwer pan faster solution key equation decoding bch codes proceedings annual acm symposium theory computing ser stoc new york usa acm schnelle multiplikation von polynomen der charakteristik acta informatica vol online available http cantor kaltofen fast multiplication polynomials arbitrary algebras acta informatica vol gathen gerhard modern computer algebra new york usa cambridge university press gao mateer additive fast fourier transforms finite fields ieee trans inf theory vol dec lin chung han novel polynomial basis application erasure codes foundations computer science focs ieee annual symposium oct ore special class polynomials trans amer math vol nov cantor arithmetical algorithms finite fields journal combinatorial theory series vol von zur gathen gerhard arithmetic factorization polynomial proceedings international symposium symbolic algebraic computation zurich switzerland mateer fast fourier transform algorithms applications dissertation clemson usa moenck fast computation gcds acm symposium theory computing stoc shiozaki decoding redundant residue polynomial codes using euclid algorithm ieee trans inf theory vol sep rockliff codes online available http ppendix olynomial muliplication formal derivative new basis showed polynomial multiplication formal derivative take similar procedure show corresponding operations muliplication multiply two polynomials exists fast approach based fft techniques approach also applied finite fields let basis overp denote two polynomials product mod computed ifftx fftx fftx vector represents coefficients degree similarly defined accordingly operation performs pairwise multiplication two vectors requires one point ifft two ffts multiplications thus complexity formal derivative polynomial formal derivative given theorem constant computed recursively let recursive form complexity written ppendix roof emmas proof lemma proof summation two terms first term proof lemma deg proof definition reformulated deg deg deg deg deg completes proof proof lemma proof proof follows mathematical induction based case shows following holds assume holds deg second term deg deg deg theorem given deg deg deg deg multiplied get completes proof proof lemma proof deg deg deg rewritten moreover deg deg deg degree term deg concludes completes proof deg deg first term reformulated deg similar step shown second stem formulated thus deg equation rewritten completes proof deg multiply obtain proof lemma proof based case deg see algorithm line clear holds assume algorithm valid hgcd degree deg case divided three individual polynomials expressed line hgcd called obtain possesses divided get multiplying obtain degree term deg follows deg deg thus deg completes proof becomes proof lemma proof two terms recalled deg let equivalent deg note computed line shows satisfies equality thus return line valid line divided get degrees deg deg matrix form reformulated decomposed several polynomials proof lemma proof assume hgcd valid recursive calls line line valid assume clear call line satisfies condition since high degree portions respectively call line first consider degree simplicity denoted deg deg deg assumption similarly rmll rmm deg max deg rmm rmh deg deg deg deg deg gives deg max multiplying obtain rmm adding side rmll rmll rmll rmm thus inequality deg deg line valid line rmm quotients dividing deg rmm deg due condition line equivalent hence deg deg deg line calls hgcd rmm obtain possessing rmm deg rmll substituting obtain rmll assumption rmm treated quotient dividing implies deg rmm deg line implies deg requirements call line verified proof lemma proof algorithm three returns lines assume recursive call hgcd line line outputs valid results line clear see line show degree least deg deg rmll assumption deg deg return results line verifies let consider line denoted condition line first condition holds let consider line denoted objective prove degree least deg deg deg deg deg deg deg deg max deg deg deg deg deg deg deg assumptions deg deg deg deg deg deg deg deg verify reformed max assumptions deg deg rmll deg verifies deg deg deg deg deg deg deg deg deg deg assumptions deg deg deg deg verified verification considered follows reformed deg deg deg deg deg summed resulting thus verified proof lemma proof proof follows mathematical induction pick base case high degree part divided two equal written verifies true show reformed deg deg deg deg deg deg deg deg deg assumptions line algorithm gives deg deg deg deg deg deg deg deg deg based assumptions seen degrees elements determined second term precisely proof lemma proof assume recursive call hgcd line line outputs valid results base case line clear condition holds notice deg special case treat deg case line objective prove deg deg deg deg deg deg becomes assumptions line algorithm gives deg rmll deg deg algorithm line computes line computes vector calculated line computed line line requires pointwise additions written vector addition assume holds thus becomes extract computations regarding similarly decomposes ifft two point iffts resulting completes proof deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg deg
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modeling control synchronous generators converter faults christoph urs korbinian abstract feb mathematical modeling faults converters current control isotropic synchronous generators discussed proposed converter model generic fault independent operation mode electrical machine proposed current control system gives improved control performance reduced torque ripple faults modifying strategy adapting modulation scheme iii injecting additional reference currents theoretical derivations model control validated comparative simulation measurement results index terms synchronous generator current control control fault control modulation injection wind turbine systems statement paper submitted publication ieee transactions power electronics ontents introduction modelling model pmsm model converter model converter without faults model converter one fault iii current control system standard control system control controllers feedforward compensation reference voltage saturation modulation control performance standard control system fault phase proposed control system modified control extension strategy modification svm modulation injection optimal implementation experimental verification experimental setup implementation discussion experiments experiment experiment experiment conclusion hackl schechner research group control renewable energy systems cres munich school engineering mse technische tum germany pecha institute electrical energy conversion university stuttgart germany authors alphabetical order contributed equally paper corresponding author hackl references otation natural real complex numbers real numbers greater equal real imaginary part column vector mean transposed defined diag diagonal matrix entries diag identity matrix zero matrix equivalence follows invoking physical quantity elements mod remainder division extension inverse tangent function whole circle clarke inverse clarke sin transformation matrix sin cos park transformation matrix rotation matrix ntroduction faults converters electric drives gained increasing attention last years especially detection faults converter identification faulty switch focus research various detection methods presented hence fault detection topic paper particular wind turbine systems desirable continue operation even presence faults converter however faulty converter causes increased losses oscillations torque harm mechanical components generator wind turbine analyze impact faults model faulty converter proposed model determines phase voltages electric machine connected faulty converter using called pole voltages converter fault one switching devices deviation pole voltage respective phase occurs affects three phase voltages pole voltages however rather unintuitive quantity compared example switching state power electronic devices phase voltages machine moreover use pole voltages simulations makes converter model unnecessarily complicated ensure safe uninterrupted operation electrical drive control strategy implemented modified space vector modulation svm proposed converters contributions adapt switching patterns replace space vectors applied due fault however papers neither consider optimal phase shift angles applied voltage current vector adapt current controllers operation although shown later measures additionally significantly improve overall control performance converters using npc configurations redundancy switching states used compensate infeasible switching states generate desired voltage output nevertheless see addition proposes inject additional current shift phase angle reference voltage current fault occurs one outer switches helps avoid infeasible switching states therefore reduces current distortion faulty phase generator however use redundant switching vectors used converters since sufficient redundancy available switching states impact open switch fault reduces feasible voltage area voltage hexagon converter significantly moreover current control system impact control performance faults discussed detail proposes consider converters faults rectifiers upper lower switches assumed simply diodes investigates possible avoidance infeasible zero switching vectors space vector modulation svm achieve minimal current distortion phase shift current voltage vector proposed phase shift achieved injecting appropriate investigations paper show considered pmsm optimal phase shift angle guarantee minimal current distortion moreover impact fault control performance current controller windup effects neither addressed tackled generic converter model covering faults simulation purposes provided another possibility ensure continued operation electric machine use converter configurations different topologies methods control faulty converters described example case faults fourth inverter leg used neutral point machine connected midpoint bus solutions compensate loss phase faulty switch however additional costly hardware components reconfigurations required contrast standard converter configurations paper current control system wind turbine systems proposed machineside converter exhibits faults proposed control system require hardware modifications easy implement since required extensions standard control system straightforward furthermore generic mathematical phase model faulty converter proposed model relies switching states upper lower switches determines phase voltages generator depending sign direction current faulty phase therefore model implemented quickly efficiently simple easy understand allows precise simulations give almost identical results conducted experiments laboratory based precise simple model impacts fault current control performance standard control system analyzed control system proposed proposed control system combines different modifications like improved strategy modified svm iii optimal current reference bringing phase angle voltage current generator optimal value neither considered machine modifications positive effects control performance faults discussed detail effectiveness proposed modifications finally illustrated validated comparative simulation measurement results odelling section models synchronous machine pmsm faults introduced models derived generic frame model pmsm stator voltages isotropic pmsm given example uabc iabc abc abc stator voltage vector uabc uas ubs ucs stator resistance stator current vector iabc stator flux linkage vector abc note stator phase currents sum zero ias ibs ics due star connection stator windings stator flux linkage abc abc abc iabc cos cos cos abc vas stator main inductance stator leakage inductance depends inductance matrix labc abc abc vas stator currents flux linkage vector linkage amplitude number pole pairs machine mechanical angle rad initial angle rad permanent magnet inserting yields current dynamics combined mechanical dynamics frame see fig follows sin abc abc abc abc sin sin abc abc abc rad initial currents iabc initial angular velocity initial machine angle abc rad induced voltage vector total inertia drive train machine torque turbine torque machine torque computed follows example sin abc abc abc abc ics sin sin remark field orientation aligning synchronously rotating frame linkage applying clarke park transformation xabc machine converter converter link electrical machine udc cdc fig converter synchronous machine dynamics machine torque yields machine dynamics abc iks linkage orientation simply field orientation see chapter sect stator voltages uks uds uqs stator currents iks ids iqs stator inductance vas flux linkage machine torque iks iks iqs model converter model converter without faults figure shows converter synchronous machine output voltage converter stator voltage given abc abc udc uabc depends fault free case symmetrical electrical machine actual switching vector means upper switch sas sbs scs actual voltage udc switching vector sabc sabc closed represents closed lower switch example switching vector sabc yields closed switches abc open applying clarke transformation stator allows transform abc fixed frame follows uss model converter one fault first model derived fault afterwards generalization faulty converter model presented without loss generality fault assumed appear switch hence switch always open independent switching vector sabc converter solely depend voltage switching vector sabc fault present voltage uabc also sign direction current phase figure illustrates different connection possibilities windings electrical machine taking sign current account whether diode conducting compare also fig resulting shifted voltage hexagon case shown fig following observations made voltage vectors see blue symbols shifted udc negative see ias ias fig normal operation feasible voltage vectors see magenta symbols shifted udc negative see ias ias fig normal operation feasible voltage vectors see green symbols shifted see fig normal operation feasible fig feasible voltage areas voltage hexagon feasible voltage vectors uss faults shown depending direction current ias ias full voltage hexagon used ias ias feasible areas voltage hexagon become smaller critical case occurs ias sectors iii feasible combining observations converter model must extended fault ias uas udc ubs ias ias ias ucs ias ubs ias udc ias uas ucs ias ias ias ias ias ias uas udc ias ubs ias ucs ias ias abc electrical equivalent circuit abc voltage hexagon shifted voltage vectors possible fig illustration impact fault electrical equivalent circuit converter pmsm simplicity windings shown inductances voltage hexagon shifted voltage vectors ias udc ias ias iii udc fig voltage hexagon feasible sectors voltage vectors fault depending current follows uabc abc switching matrix ias exclusively models converter fault changes direction phase current ias generalizing observations arbitrary fault one six switches introducing negated switching vector sabc sabc table complete model faulty converter switching matrices faulty switches resp abc abc converter output phase voltage faults upper switches abc abc converter output phase voltage faults lower switches leads different switching matrices ixs ixs respectively finally switching matrices possible faults six switches respective phase current directions collected table details omitted due space limitations iii urrent ontrol ystem controllers control common choice current control system electrical drives see sect sect cases controllers equipped strategy feedforward compensation terms compensate section first standard control briefly revisited afterwards crucial modifications improve control performance faults proposed impact fault standard control system improvements achieved proposed control system illustrated analyzed simulations see sect sect finally simulation results validated comparative measurement results sect standard control system control fig block diagram standard control system consisting controllers anti feedforward compensation reference voltage saturation modulation depicted following subsections block explained briefly controllers controllers weight integrate current control error ekis eqis iis ref ref defined difference reference currents coming outer control loops actual currents output uks uds uqs dynamics integrator controllers decision function faw follows uks ekis ref proportional integral controller gains kpd kpq kid kiq respectively example tuning according magnitude optimum see leads following controller gains kpd kpq kid kiq fsw switching frequency reasonable tuning rule might also applicable decision function ref udc faw ref else compensation controllers comp eis ref modulation saturation ref ref ref ref sat abc eis ref ref comp ref ref funct fig standard control system control controllers feedforward compensation reference voltage saturation enables disables integration integral control action conditional integration details see sect applied reference voltage vector amplitude ref defined uks uks comp ref uss ref ref ref ref ref ref exceeds maximally available voltage amplitude given udc sin cos udc mod ref ref within full voltage hexagon see fig maximally available amplitude converter depends voltage reference angle rad voltage note varies inside voltage hexagon larger maximum voltage amplitude udc udc minimum voltage amplitude udc udc invoking trigonometric identities leads expression considering first sector voltage hexagon note first sector coincides using auxiliary phase angle defined generalized formula available amplitude obtained sectors voltage hexagon feedforward compensation compensation terms current dynamics realized following feedforward control action uks comp uus comp comp least steady state cancels influence dynamics vice versa sect reference voltage saturation avoid undesirable infeasible output voltages converter applying physically infeasible voltage reference vectors computed reference voltage vector uss ref saturated necessary follows ref ref ref sat cos sin else note reference voltage saturation alter length voltage vector direction modulation generate switching sequence particular switching vector sabc based saturated reference voltage uss ref sat symmetrical space vector modulation svm used paper chap sect boundary adjacent space vectors respective sector usually zero vectors applied approximate reference voltage vector uss ref sat one switching period tsw tsw divided three time intervals first vector second vector zero vectors tsw see fig fig voltage reference vector saturated sect implementation negative time cause strange behaviour svm control performance standard control system fault phase measure evaluate control performance standard control systems total harmonic distortion thd used total harmonic distortion thdias phase current ias computed follows thdias ina rms fundamental harmonic current component respectively figure shows control performance standard control system fig fault phase upper subplot illustrates reference ids ref iqs ref actuals currents ids iqs frame whereas lower subplot shows phase currents ias ibs ics time clearly seen current ias phase sinusoidal negative close zero positive deviation leads usually expected currents ids iqs significantly differ respective reference values ids ref iqs ref almost time current iqs tends zero positive ias moreover even negative correct ias current iqs capable tracking reference iqs ref particular nonlinearly oscillating evolution iqs leads noticeable torque ripples current ids contribute torque recall increases copper losses machine conclusion standard control system performance acceptable must improved allow safe uninterrupted operation wind turbine system proposed control system modified control illustrated fig standard control system performance poor acceptable fault present without altering hardware principle control system three software modification proposed improve control performance wind turbine system faults one phase three modifications extension strategy modification svm torque ripple minimization injecting optimal modification explained detail positive effect control performance control system illustrated simulation results later sect modifications implemented laboratory test bench effectiveness validated measurements block diagram improved control system depicted fig modifications highlighted blue extension strategy improve control performance faults first step strategy current controllers modified overshoots fault recall fig least windup integral control action controllers positive almost zero ias avoid windup additional condition considering current direction see fig must introduced leads extended decision function ref udc ias faw ref else faults phase replaces faw constant represents maximally admissible current chosen negative account chattering phase current ias around zero recall fig note faulty phase fault respective phase current direction ibs ics must considered instead ias fig improved control system performance due extended strategy shown abccurrents lower subplot alter much almost improvement visible thd reduces slightly thdias tracking performance currents ids particular iqs improved substantially positive almost zero ias currents still perfectly track references negative ias currents capable almost asymptotic reference tracking especially current ripple drastically reduced negative ias root mean square rms value defined fundamental period current compensation controllers comp injection eis ref ref modulation saturation ref ref sat abc eis ref ref ref comp ref ref extended decision function fig control system modified control changes blue fault phase controllers improved feedforward compensation reference voltage saturation sas sas sbs tsw tsw sbs tsw tsw tsw tsw scs tsw scs tsw standard svm modulation ref fig switching patterns generate voltage reference vector sector iii linear combination space vectors zero vectors zero vector modification svm modulation second step improve control performance faults modification modulation note fault one upper switches leads shifted zero vector ias see fig whereas fault one lower switches shifts zero vector hence one zero vectors zero using modulation allows use zero vector chap example fault zero vector applied see fig faults zero vector used fig positive effect modulation control performance thd ias illustrated iss iib iia iss uss ref iii uss ref iib iia iii uss ref uss ref iss iss fig stator current vector voltage reference vector ref time upper subplots show currents references whereas lower subplot depicts modified control system extended modified svm clearly intervals ias significantly shorter moreover positive almost sinusoidal ias currents feasible due modified svm thd value drastically reduced thdias tracking control performances currents ids iqs improved well injection optimal last improvement inject optimal additional minimize thd faulty phase even following fault considered faults modifications straight forward discussed sect fault ias output voltages provided faulty converter limited ias voltage vectors sectors iii feasible see fig principle idea optimal injection generate auxiliary reference voltage vectors within two feasible sectors long possible goal determine optimal phase shift rad stator current iss reference voltage uss ref illustrate idea fig phase shifts shown two time instants iss located negative positive respectively note phase current ias stator current space vector iss located right see fig precisely ias ias becomes positive thereafter iss lies negative whereas ias ias becomes negative afterwards iss aligned positive clearly within interval current moves optimally corresponding stator voltage reference space vector uss ref within sectors iii long possible order apply feasible correct voltages generator however two sectors represent span possible apply correct voltages whole ias fault remaining incorrect voltages applied faulty converter affect shape currents cause deviations desired sinusoidal waveform depending phase shift stator current reference voltage different parts ias affected fault see fig uss ref starts sector time first current incorrect voltages applied generator soon uss ref enters sector iii correct voltages provided even faulty converter uss ref sectors iii correct voltages give rise sinusoidal current see fig behaviour flipped time uss ref starts already feasible sector iii hence first current correct voltages applied soon uss ref enters sector incorrect voltages generated converter rest time concluding order fully benefit two feasible sectors correct voltages generated ias phase shift must within interval observations also validated simulation results presented fig see fig first phase current ias jitters around zero contrast last desired sinusoidal characteristic achieved see fig phase current ias exhibits sinusoidal characteristic first whereas last deteriorated cur cur thdia thdia abc fig currents references different phase shifts remark converter outputting pmsms motor mode inverters wind turbine systems solely phase shifts feasible hence possible benefit sectors iii phase shift altered injection also changes ratio active power reactive power var since tan uks iks uks iks see ids moreover note due chosen independently desired torque order derive analytical expression reference current ids ref current ids following assumption imposed assumption copper losses pmsm negligible current dynamics steady state solving steady state uks inserting result gives ids iqs ids ids iqs iqs tan polynomial ids root smaller amplitude used iqs tan ref tan tan tan iqs tan tan tan tan tan hence large machines reference current ids ref depends machine parameters current iqs reference iqs ref desired phase angle exists optimal value minimize thd phase current ias considered machine optimal value opt found iterative simulations results depicted fig clearly machines optimal value might different finally fig simulation results overall control system extended modified svm modulation optimally injected shown upper subplot depicts currents ids iqs reference values whereas lower subplot illustrates shape thd value scenario thdias clearly lowest compared simulation results figures solution front root would lead higher current magnitude opt thdia different phase angles tracking optimal phase angle opt actual phase angle fig thdias phase current different phase angles tracking optimal phase angle opt fault phase moreover reference tracking capability particular iqs best implies torque also minimized recall leading less stress mechanical drive train fig tracking optimal phase angle opt actual phase angle shown due remaining time periods correct voltage vector applied phase current close zero optimal phase angle achieved time remark possible limitation injection wind turbine systems due current converter additional injection might feasible rated torque isotropic generator wind turbine systems therefore uninterrupted operation wind turbine system faults might possible wind speeds unless pitch control system incorporated control system turbine torque proportional wind speed pitch angle must decreased changing pitch angle current rating converter exceeded details see chap mplementation experimental verification section implementation experimental validation comparison simulation measurement results discussed three experiments laboratory conducted validate accuracy proposed mathematical model converter fault real electrical drive system fault verify effectiveness proposed modifications extension strategy modulation injection optimal generator control performance compare simulation experimental results illustrate impact occurring fault interval positive effect proposed modification control performance laboratory electrical drive system experimental setup implementation measurements conducted laboratory test bench depicted fig anisotropic reluctance synchronous machine rsm underlying nonlinear current controllers isotropic permanentmagnet synchronous machine used generator described previous sections drives controlled dspace system applies switching signals switching vectors respective converters connected pmsm converter modified upper lower switch addressed individually allows emulate faults experiments without loss generality faults phase considered simulated emulated implementation simulations measurements performed using fig block diagram implementation shown parameters laboratory setup listed tab coincide used simulations note measured currents filtered analogue filter converter sampled switching frequency whereas simulated currents filtered discussion experiments experiment simulation measurement scenario experiment follows converter emulates fault standard control system described sect implemented simulation note produced generator torque wind turbine systems equal reference value wind turbine efficiency power production reduced cur cur standard control performance thdia improved control performance particular extended thdia due cur cur improved control performance due extended modified svm modulation thdia improved control performance due extended antiwindup modified svm modulation injection thdia fig comparative control system performance fault phase fig laboratory test bench dspace system inverters connected reluctance synchronous machine rsm synchronous machine pmsm torque sensor table simulation measurement data description symbol value unit udc fsw khz simulation parameters sampling time converter voltage switching frequency synchronous machine generator isotropic stator resistance stator inductance linkage number pole pairs machine inertia reluctance synchronous machine anisotropic stator resistance stator inductances nonlinear see fig number pole pairs machine inertia current control system pmsm controller gains avs maximum current implementation control system dspace system current controllers see fig fig udc ref comp ref ref ref sat abc abc ref sat abc abc pmsm abc converter abc modulation udc system model laboratory hardware fig block diagram implementation svm converter fault pmsm current control system dspace system measurement results experiment shown fig measured quantities labeled additional subscript meas obviously simulation measurement results match closely hence proposed mathematical model valid allows simulate behavior real system precisely note due smaller power rating rsm speed controller rsm capable compensate large ripples induced faulty experiment experiment fault phase pmsm emulated time control system extended control proposed sect extended modulation optimal ids implemented simulation measurement figure shows comparative simulation measurement results simulation measurement results match closely moreover also thdias thdias meas almost identical conclusion proposed modifications also effective real world outcomes theoretical simulative analysis sect confirmed cur cur vel vel experiment comparison simulation measurement results standard control system thdia simulation thdia meas measurement experiment comparison simulation measurement results control system thdia simulation thdia meas measurement fig comparison simulation measurement results standard control system control system extended modified svm modulation optimally injected experiment last experiment measurement scenario comprises sequence events see fig time interval standard control system without case time interval standard control system fault time interval standard control system extended fault time interval standard control system extended modified svm modulation fault time interval control system extended modified svm modulation optimal injection fault measurement results shown fig first second subplots show measured ids iqs references ids ref iqs ref phase current ias reference ias ref respectively third subplot shows rotational speed reference soon fault occurs currents ids iqs start oscillate within band phase current ias track reference since positive reproduced resulting torque ripples pmsm deteriorate speed control rsm rotational speed begins oscillate amplitude roughly rad extended strategy enabled band current speed oscillations decreases slightly phase current ias roughly track negative reference additional use modified svm modulation reduces oscillation band significantly improves current tracking capability note iqs approach zero anymore finally enabling optimal ids achieves ids track current reference iqs ref small ripples due optimal injection ids still oscillating around reference value within band resulting magnitude phase current ias increased approximately torque ripples drastically reduced speed control performance almost good case onclusion paper generic mathematical model converter faults derived output voltages faulty converter computed based switching vector voltage sign current phase fault model holds faults six switches next step impact fault current control system isotropic synchronous generators investigated faulty switch known corresponding diode still working properly three extension control system proposed improve control performance faults three modifications reduce thd cur vel case extended fault modulation optimal fig experiment measurement results current control system four different scenarios case fault iii extended additionally modulation additionally optimal injection faulty phase current torque ripples therefore stress mechanical drive train moreover safe uninterrupted operation generator guaranteed proposed modifications extension anti strategy current controllers modification svm modulation iii injection optimal modifications explicitly illustrated implemented fault phase however generic model provided descriptions allow implement modifications fault converter finally comparative simulation measurement results illustrate validate accuracy proposed model faulty converter effectiveness functionality proposed modifications laboratory test bench acknowledgement authors deeply indebted max lindner adapting laboratory test bench make measurements converter faults possible project received funding bavarian ministry education culture science art eferences sharma literature review igbt fault diagnostic protection methods power inverters ieee transactions industry applications vol sept rothenhagen fuchs performance diagnosis methods igbt open circuit faults three phase voltage source inverters variable speed drives european conference power electronics applications sept choi lee blaabjerg lee fault diagnosis control npc inverter ieee transactions power electronics vol oct lee lee blaabjerg fault detection method converter using npc topology wind turbine systems ieee transactions industry applications vol jan choi lee blaabjerg diagnosis tolerant strategy fault inverter systems ieee transactions industry applications vol jan choi jeong lee blaabjerg method detecting fault npc inverter system ieee transactions power electronics vol june kim kim lee lee diagnosis methods igbt open switch fault applied pwm converter journal power electronics vol freire estima cardoso fault diagnosis pmsg drives wind turbine applications ieee transactions industrial electronics vol sept eickhoff seebacher muetze strangas enhanced fast detection open switch faults inverters electric drives ieee transactions industry applications vol araujo ribeiro jacobina silva lima fault detection damage pwm motor drive systems ieee transactions power electronics vol mar kim choi jung fault model performance evaluation permanent magnet synchronous motor winding shorted turn inverter switch open iet electric power applications vol april jung park kim cho youn diagnosis fault pwm inverters synchronous motor drive systems ieee transactions power electronics vol may kim lee lee diagnosis control pwm converter systems ieee transactions industry applications vol july sobanski analysis space vector modulation technique induction motor drive international power electronics motion control conference exposition sept moon kim lee lee fault tolerant control strategy pwm converter multiple faults conditions annual ieee applied power electronics conference exposition apec feb jung lee kim operation fault pwm converter international power electronics motion control conference exposition sept lee lee fault detection method tolerance controls based svm rectifier unity power factor ieee transactions industrial electronics vol dec lee lee fault tolerance control rectifier wind turbine systems ieee transactions industrial electronics vol feb freire permanent magnet synchronous generator drives wind turbine applications phd thesis university coimbra gaeta scelba consoli modeling control pmsms fault ieee transactions industry applications vol jan wallmark harnefors carlson operation inverters pmsm drives european conference power electronics applications sept bolognani zordan zigliotto experimental control pmsm drive ieee transactions industrial electronics vol oct hackl based adaptive control mechatronics theory application lecture notes control information sciences berlin springer international publishing doncker pulle veltman advanced electrical drives power systems berlin dirscherl hackl schechner modellierung und regelung von modernen windkraftanlagen eine see https english translation elektrische antriebe regelung von antriebssystemen elektrische antriebe regelung von antriebssystemen berlin heidelberg shmilovitz definition total harmonic distortion effect measurement interpretation ieee transactions power delivery vol jan valentine motor control electronics handbook handbooks companies hackl schechner feedforward torque control wind turbines impacts annual energy production gross earnings journal physics conference series vol burton sharpe jenkins bossanyi wind energy handbook john wiley sons hackl kamper kullick mitchell current control reluctance synchronous machines online adjustment controller parameters proceedings ieee international symposium industrial electronics isie santa clara usa institute electrical electronics engineers ieee jun eldeeb hackl horlbeck kullick analytical solutions optimal reference currents mtpv mtpf control anisotropic synchronous machines proceedings ieee international electric machines drives conference iemdc miami usa
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universality laws randomized dimension reduction applications sep samet oymak joel tropp bstract dimension reduction process embedding data lower dimensional space facilitate analysis euclidean setting one fundamental technique dimension reduction apply random linear map data dimension reduction procedure succeeds preserves certain geometric features set question large embedding dimension must ensure randomized dimension reduction succeeds high probability paper studies natural family randomized dimension reduction maps large class data sets proves phase transition success probability dimension reduction map embedding dimension increases given data set location phase transition maps family furthermore map stability properties quantified restricted minimum singular value results viewed new universality laws stochastic geometry universality laws randomized dimension reduction many applications applied mathematics signal processing statistics yield design principles numerical linear algebra algorithms compressed sensing measurement ensembles random linear codes furthermore results implications performance statistical estimation methods large class random experimental designs date november revised august september september mathematics subject classification primary secondary key words phrases conic geometry convex geometry dimension reduction invariance principle limit theorem random code random matrix randomized numerical linear algebra signal reconstruction statistical estimation stochastic geometry universality oymak tropp ontents overview universality phenomenon randomized dimension reduction technical setting phase transition uniformly random partial isometries types random linear maps universality law embedding dimension universality law restricted minimum singular value applications universality scope universality phenomenon reproducible research roadmap notation part main results random matrix models bounded random matrix model random matrix model universality law embedding dimension embedding dimension problem formulation concepts spherical geometry statistical dimension functional theorem universality embedding dimension theorem proof strategy theorem extensions universality law restricted minimum singular value restricted minimum singular value problem formulation excess width functional theorem universality restricted minimum singular value theorem proof strategy theorem extensions part applications recovery structured signals random measurements phase transition sparse signal recovery signal recovery problems complements decoding structured errors decoding sparse errors demixing problems randomized numerical linear algebra subspace embeddings sketching least squares prediction error lasso sparse linear model lasso proof proposition part iii universality restricted minimum singular value proofs theorem theorem universality randomized dimension reduction restricted singular values bounded random matrix theorem main result bounded random matrix model proof theorem concentration setup proof theorem dissection index set proof theorem lower bound rsv proof theorem upper bound rsv convex set restricted singular values random matrix corollary main result random matrix model proof theorem corollary proof theorem corollary theorem concentration restricted singular values proposition lower tail rsv proposition upper tail rsv theorem probability bounds dissections dissection excess width dissection restricted singular value theorem replacing entries random matrix proof proposition proposition discretizing index set proposition smoothing minimum proposition exchanging entries random matrix theorem bounding restricted singular value excess width proof proposition proposition moment comparison inequality hybrid rsv proposition role gaussian minimax theorem proposition reducing gaussian matrix gaussian vectors proposition removing remaining part original random matrix proposition replacing coordinates missing excess width proof corollary theorem truncation proof corollary corollary truncation individual random variables part universality embedding dimension proof theorem embedding fails bounded random matrix rap functional dual condition failure theorem main result bounded random matrix model proof theorem lower tail bound proof theorem truncation dissection proof theorem lower bound rap functional embedding fails random matrix corollary main result random matrix model proof theorem corollary proof proposition application rap functional decoding structured errors theorem truncation cone theorem replacing entries random matrix proof proposition proposition discretizing index sets proposition smoothing minimum proposition exchanging entries random matrix theorem bounding rap functional excess width iii oymak tropp proof proposition proposition duality rap functional proposition reducing gaussian matrix gaussian vectors proposition finding gaussian width part back matter appendix tools gaussian analysis concentration gaussian lipschitz functions gaussian minimax theorem appendix spectral bounds random matrices acknowledgments references ist igures geometry random linear map universality embedding dimension geometry restricted minimum singular value universality restricted minimum singular value restricted minimum singular value restricted maximum singular value universality recovery phase transition universality decoding phase transition universality lasso prediction error universality randomized dimension reduction verview niversality henomenon paper concerns fundamental question stochastic geometry likely random subspace fixed dimension intersect given set problem roots earliest research spherical integral geometry also arises asymptotic convex geometry recent years question attracted fresh attention central analysis randomized dimension reduction paper establishes striking universality phenomenon takes place stochastic geometry problem given set answer question essentially every distribution random subspaces induced natural model random linear maps universality also manifests metric variants ask far random subspace lies set discuss implications results geometry random matrix theory numerical analysis optimization statistics signal processing beyond randomized dimension reduction dimension reduction operation mapping set large space smaller space ideally action distills information set allows develop efficient algorithms processing information setting euclidean spaces fundamental method dimension reduction apply random linear map point set important random linear map preserve geometric features set particular want linear map map point set origin equivalently null space random linear map intersect set see emerges naturally context randomized dimension reduction technical setting let introduce framework study problem natural treat question spherical geometry scale invariant fix ambient dimension consider closed subset euclidean unit sphere moment also assume spherically convex intersection convex unit sphere construct random linear map embedding dimension exceed ambient dimension vary distribution random linear map map null induces different distributions subspaces codimension may reformulate language given embedding dimension probability null equivalently probability say random projection succeeds conversely say random projection fails see figure illustration intuition fixed choice projection likely succeed embedding dimension increases furthermore random linear map fixed embedding dimension less likely succeed size set increases justify heuristics complete detail phase transition uniformly random partial isometries begin case literature already contains comprehensive answer question natural type random embedding uniformly random partial isometry partial whose null space null drawn uniformly random haar measure grassmann manifold subspaces codimension invariance properties distribution allow complete analysis action spherically convex set chap recent research shown convert complicated exact formulas interpretable results convex cone convex set satisfies partial isometry satisfies condition transpose operation identity map oymak tropp range range null null igure geometry random linear map identify linear map orthogonal projector whose range orthogonal complement null diagram random orthogonal projector applied closed spherically convex set likelihood given configuration depends statistical dimension set eft success regime null space null intersect image contain origin ight failure regime null space null intersects set image contains origin modern theory expressed terms geometric functional called statistical dimension max normal statistical dimension increasing respect set inclusion values range zero empty set whole sphere furthermore functional computed accurately many cases interest see section details statistical dimension demarcates phase transition behavior uniformly random partial isometry embedding dimension varies closed spherically convex set results thm prop demonstrate implies high probability implies high probability number positive universal constant terms uniformly random projection spherically convex set likely succeed precisely embedding dimension larger statistical dimension see figure plot exact probability uniformly random partial isometry annihilates point specific set remark related work results thm thm contain good bounds probabilities probabilities approximated precisely introducing second geometric functional estimates depend spherical crofton formula eqns gives exact probabilities less interpretable form related phase transition results also obtained via gaussian minimax theorem see cor rem thm see results uniformly random partial isometries types random linear maps research outlined section delivers complete account uniformly random partial isometry behaves presence convexity universality randomized dimension reduction probability success random embedding nonnegative unit vectors probability success uniformly random theory rademacher empirical nonzero rademacher empirical student dof empirical embedding dimension igure universality embedding dimension plot describes behavior four types random linear maps applied set unit vectors nonnegative coordinates dashed line marks statistical dimension set gray curve interpolates exact probability uniformly random partial isometry succeeds function embedding dimension markers indicate empirical probability trials dimension reduction succeeds random linear map specified distribution see sections details contrast literature contains almost precise information behavior random linear maps nevertheless applications may work types random linear maps motivating example many algorithms numerical linear algebra depend randomized dimension reduction context uniformly random partial isometries expensive construct store perform arithmetic appealing implement random linear map discrete sparse structured lack detailed theoretical information linear maps behave makes difficult design numerical methods guaranteed performance however use computation investigate behavior types random linear maps figure presents results following experiment consider set unit vectors nonnegative coordinates according statistical dimension formula tells expect phase transition behavior uniformly random partial isometry embedding dimension using eqn compute exact probability uniformly random partial isometry succeeds function baseline compare empirical probability trials random linear map independent entries succeeds also display experiments nonzero rademacher linear map linear map student entries see section details experiment discover three linear maps act almost exactly way uniformly random partial isometry universality phenomenon remarkable four rademacher random variable takes two values equal probability oymak tropp null range igure geometry restricted minimum singular value identify partial unitary linear map orthogonal projector whose range orthogonal complement null diagram orthogonal projector applied compact convex set restricted minimum singular value distance origin image linear maps rather different distributions present literature contains information phenomenon occurs universality law embedding dimension central goal paper show substantial class random linear maps phase transition embedding dimension universal rough statement main result let closed spherically convex set suppose entries matrix random linear map independent modest amount particular may consider random linear maps arbitrarily small constant proportion nonzero entries class random linear maps demonstrate implies high probability implies high probability notation suppresses constants depend regularity random variables populate see theorem section complete statement result states random linear map likely succeed spherically convex set precisely embedding dimension exceeds statistical dimension set learn phase transition embedding dimension universal class random linear maps provided small compared ambient dimension note random linear map standard normal entries behavior uniformly random partial isometry almost surely null space uniformly random subspace codimension analysis explains dominant features experiment figure universality law restricted minimum singular value also matter significant interest understand stability randomized dimension reduction quantify stability random linear map compact convex set using restricted minimum singular value min standardized random variable mean zero variance one symmetric random variable distribution negation concreteness may assume entries five uniformly bounded moments universality randomized dimension reduction restricted minimum singular value restricted minimum singular value random embedding probability simplex width asymptotic theory rademacher empirical nonzero rademacher empirical student dof empirical embedding dimension igure universality restricted minimum singular value plot describes behavior three types random linear maps applied probability simplex dashed line marks minimum dimension uniformly set likely succeed gray curve interpolates value positive part width obtained asymptotic calculation markers give empirical estimate trials restricted minimum singular value random linear map drawn specified distribution see section information restricted minimum singular value large random image far origin embedding stable deform either linear map set still sure embedding succeeds restricted minimum singular value small random dimension reduction map unstable zero random dimension reduction map fails see figure diagram second major theorem universality law restricted minimum singular values random linear map result expressed using geometric functional called width min normal width increases parameter decreases respect set inclusion typical scale addition excess width evaluated precisely many situations interest see section details suppose entries matrix independent standardized symmetric modest amount regularity compact convex subset unit ball establish high probability notation suppresses constants depend regularity random variables theorem section contains detailed statement summary provided set small restricted minimum singular value depends primarily geometry set embedding dimension rather distribution random linear map see figure numerical illustration fact applications universality randomized dimension reduction generally random matrices become ubiquitous information sciences consequence universality laws outlined section wide range implications oymak tropp signal processing main idea field compressed sensing acquire information structured signal taking random linear measurements literature contains extensive empirical evidence many types random measurements behave indistinguishable fashion work gives first explanation phenomenon section stochastic geometry results also indicate facial structure convex hull independent random vectors drawn appropriate class depend heavily distribution section coding theory random linear codes provide efficient way protect transmissions error demonstrate class random codes resilient sparse corruptions number errors corrected depend specific choice codebook section numerical analysis research provides engineering design principle numerical algorithms based randomized dimension reduction select random linear map favorable implementation still confident detailed behavior algorithm approach allows develop efficient numerical methods also rigorous performance guarantees section random matrix theory work leads new proof law minimum singular value random matrix independent entries section statistics lasso widely used method performing regression variable selection demonstrate prediction error associated lasso estimator universal across large class random designs statistical error models also show lad regression correct small number arbitrary statistical errors wide class random designs section remark neuroscience universality laws may even broader scientific significance conjectured experimental evidence brain may use dimension reduction compress information universality laws suggest many types uncoordinated random activity lead dimension reduction methods behavior result indicates hypothesis neural dimensionality reduction may biologically plausible scope universality phenomenon universality phenomenon developed paper extends beyond results establish apparently related problems universality hold let say words examples first appear important random linear map independent entries extensive evidence structured random linear maps also universality properties example see second restricted minimum singular value type functional universality visible instance suppose convex lipschitz function consider quantity min optimization problems form play central role contemporary statistics machine learning likely value optimization problem universal wide class random linear maps furthermore believe techniques adapted address question hand geometric functionals involving norms need exhibit universality consider restricted minimum singular value min nontrivial sets value optimization problem varies lot choice random linear map instance let first standard basis vector define shifted euclidean ball universality randomized dimension reduction restricted minimum singular value random embedding shifted euclidean ball rademacher empirical gaussian empirical student dof empirical nonzero rademacher empirical restricted minimum singular value embedding dimension igure restricted minimum singular value plot describes behavior four types random linear maps applied set first standard basis vector dashed line stands phase transition embedding dimension set markers give empirical estimate trials restricted minimum singular value mint random linear map specified distribution function embedding dimension see section details using theorem calculation sec statistical dimension circular cone verify universal phase transition successful embedding set embedding dimension result implies minimum restricted singular value also takes universal value time figure illustrates functional universal set finally functionals involving maximization necessarily exhibit universality restricted maximum singular value defined max hard produce examples restricted maximum singular value depends choice random matrix instance figure demonstrates random linear map substantial impact maximum singular value restricted probability simplex observation may surprise researchers random matrix theory ordinary maximum singular value universal class random matrices independent entries thm reproducible research paper accompanied matlab code reproduces figure stored data software also repeat numerical experiments obtain new instances figure modifying parameters code reader may explore changes affect universality phenomenon omit meticulous descriptions numerical experiments text recitations tiresome reader code offers superior documentation roadmap paper divided five parts part offers complete presentation universality laws comments proofs prospects research part outlines applications universality several disciplines contains empirical confirmation analysis part iii presents proof restricted minimum singular value exhibits universal oymak tropp restricted maximum singular value restricted maximum singular value random embedding probability simplex student dof empirical nonzero rademacher empirical gaussian empirical rademacher empirical embedding dimension igure restricted maximum singular value plot describes behavior four types random linear maps applied probability simplex markers give empirical estimate trials restricted maximum singular value random linear map specified distribution see section details behavior argument also yields condition randomized embedding likely succeed part contains proof condition randomized embedding likely fail finally part includes background results acknowledgments list works cited notation let summarize notation use italic lowercase letters example scalars boldface lowercase letters vectors boldface uppercase letters matrices uppercase italic letters may denote scalars sets random variables depending context roman letters denote universal constants may change appearance appearance sometimes delineate specific constant values subscripts crad given vector set indices write vector restricted indices particular coordinate vector given matrix sets row column indices write submatrix indexed particular component position single index always refers column submatrix indexed always work real euclidean space symbol unit ball unit sphere unadorned norm refers norm vector operator norm matrix use notation standard inner product vectors length write transpose vector matrix real number define functions max max functions bind powers square positive part symbols var refer expectation variance random variable returns probability event use convention powers bind expectation returns expectation square write indicator random variable event standardized random variable mean zero variance one symmetric random variable distribution negation reserve letter standard normal random variable boldface letters always standard normal vectors standard normal matrix dimensions determined context universality randomized dimension reduction part main results part paper introduces two new universality laws one phase transition embedding dimension second one restricted minimum singular values random linear map also include remarks proofs postpone details parts iii section introduce two models random linear maps use throughout paper section presents universality result embedding dimension section presents result restricted singular values andom atrix odels begin present two models random linear maps arise study universality one model includes bounded random matrices independent entries second allows random matrices entries bounded random matrix model first model contains matrices whose entries uniformly bounded model useful applications plays central role proofs universality results model bounded matrix model fix parameter random matrix model following properties independence entries stochastically independent random variables standardization entry mean zero variance one symmetry entry symmetric distribution boundedness entry matrix uniformly bounded identical distribution entries required cases note symmetry requirement dropped model includes several types random matrices appear frequently practice example rademacher matrices consider random matrix whose entries independent rademacher random variables type random matrix meets requirements model rademacher matrices provide simplest example random linear map appealing many applications discrete example sparse rademacher matrices let thinning parameter consider random variable distribution probability probability otherwise random matrix whose entries independent copies satisfies model random matrices useful control sparsity random matrix model next introduce general class random matrices includes examples main results concern random linear maps model model model fix parameters random matrix model following properties independence entries stochastically independent random variables standardization entry mean zero variance one symmetry entry symmetric distribution bounded moments entry uniformly bounded pth moment identical distribution entries required oymak tropp model subsumes model also encompasses many types random matrices example gaussian matrices consider random matrix whose entries independent standard normal random variables matrix satisfies requirements model contexts use gaussian random matrix study behavior uniformly random partial isometry indeed null space null standard normal matrix uniformly random subspace codimension min almost surely model contains several classes random matrices example subgaussian matrices suppose entries random matrix independent entry symmetric standardized uniformly subgaussian parameter matrices included model rademacher sparse rademacher gaussian matrices fall category example entries suppose entries random matrix independent entry symmetric standardized random variable recall real random variable density form convex normalizing constant shown thm matrices included model contrast research randomized dimension reduction allow random linear map entries heavy tails one example example student matrices suppose entry random matrix independent student random variable degrees freedom matrix also follows model finally present general construction takes matrix model produces sparse matrix also satisfies model albeit larger value parameter example sparsified random matrix let random matrix satisfies model value let thinning parameter construct new random matrix whose entries independent random variables distribution probability probability sparsified random matrix still follows model value modified value parameter niversality mbedding imension section present detailed results show large class sets embedding dimension universal large class linear maps universality randomized dimension reduction embedding dimension problem formulation let begin rigorous statement problem fix ambient dimension let nonempty compact subset contain origin let random linear map embedding dimension interested understanding probability random projection contain origin want study following predicate equivalently null say random projection succeeds property holds otherwise random projection fails goal argue large class sets large class random linear maps probability holds depends primarily choice embedding dimension appropriate measure size set particular probability depend significantly distribution random linear map concepts spherical geometry property reflect scale points set consequence appropriate translate problem question spherical geometry begin definition definition spherical retraction spherical retraction map defined every set equivalence therefore may pass spherical retraction set without loss generality obtain complete analysis random projection succeeds fails must restrict attention sets convexity property definition spherical convexity nonempty subset unit sphere spherically convex convex cone convention empty set also spherically convex particular suppose compact convex set contain origin retraction compact spherically convex next introduce polarity operation spherical sets supports crucial duality arguments definition spherical polarity let subset unit sphere polar set convention polar empty set whole sphere definition simply spherical analog polarity cones verified always closed spherically convex furthermore polarity operation involution class closed spherically convex sets statistical dimension functional intuition given compact subset probability random linear map succeeds decreases content set present context correct notion content involves geometric functional called statistical dimension oymak tropp definition statistical dimension let nonempty subset unit sphere statistical dimension defined sup addition define extend statistical dimension general subset using spherical retraction recall random vector standard normal distribution statistical dimension number striking properties include short summary see prop citations information set statistical dimension takes values range statistical dimension increasing respect set inclusion implies statistical dimension agrees linear dimension subspaces dim subspace statistical dimension interacts nicely polarity spherically convex set relation holds replace convex cone use conic polarity specific example evaluate statistical dimension nonnegative orthant indeed orthant cone also powerful machinery developed computing statistical dimension descent cone convex function finally mention statistical dimension often evaluated sampling gaussian vectors approximating expectation empirical average remark gaussian width statistical dimension related gaussian width functional bounded set gaussian width defined sup subset unit sphere inequalities see prop proof relation allows pass squared width statistical dimension gaussian width set closely related mean width sec mean width canonical functional euclidean setting sec quite right choice spherical geometry chosen work statistical dimension many geometric properties width lacks spherical setting remark convex cones papers define statistical dimension convex cone using intrinsic volumes shown statistical dimension canonical additive continuous extension linear dimension class closed convex cones sec general definition agrees original definition class universality randomized dimension reduction theorem universality embedding dimension prepared state main result probability random linear map succeeds fails given set theorem universality embedding dimension fix parameters model choose parameters number following statement holds suppose ambient dimension nonempty compact subset contain origin statistical dimension proportional ambient dimension random linear map obeys model parameters implies furthermore spherically convex implies constant depends parameter random matrix model section summarizes strategy establishing theorem detailed proof theorem appears section detailed proof theorem appears section let mention stronger probability bounds hold random linear map drawn model random linear map drawn model theorem ensures random dimension reduction likely succeed embedding dimension exceeds statistical dimension similarly theorem shows likely fail embedding dimension smaller statistical dimension provided spherically convex note interpretations require statistical dimension small compared ambient dimension already seen concrete example theorem work view calculation statistical dimension orthant theorem provides satisfying explanation figure remark prior work random linear map gaussian result theorem follows gordon cor conclusion seems recent thm knowledge literature contains precedent theorem general sets linear maps identify sporadic special cases donoho tanner studied problem recovering saturated vector random measurements via minimization work interpreted statement random embeddings set unit vectors nonnegative coordinates demonstrate phase transition embedding dimension universal rows linear map matrix independent symmetric random vectors density result actually follows classical work wendel let emphasize result apply discrete random linear maps bayati studied problem recovering sparse vector random measurements via minimization work interpreted result embedding dimension set descent directions norm sparse vector showed asymptotically phase transition embedding dimension universal result requires linear map matrix contain independent subgaussian entries absolutely continuous respect gaussian density see section discussion result also many papers asymptotic convex geometry mathematical signal processing contain theory order embedding dimension linear maps model see results allow reach conclusions existence phase transition location also extensive amount empirical work oymak tropp suggests phase transitions ubiquitous highdimensional signal processing problems theoretical explanation universality phenomenon theorem proof strategy proof theorem depends converting geometric question analytic problem first recall equivalence allows pass compact set spherical retraction next identify two analytic quantities determine whether linear map annihilates point set proposition analytic conditions embedding let nonempty closed subset unit sphere let linear map min kat implies furthermore spherically convex cone subspace min min implies finally cone subspace select arbitrary subset property cone dim subspace min min implies recall cone smallest convex cone containing set proof implication quite easy check point kat implies words second implication follows spherical duality principle suppose closed spherically convex assume cone subspace intersect must intersect see thm analytic condition min min ensures cone lies positive distance range follows intersect range duality conclude null yields finally cone subspace use dimension counting argument analytic condition implies range since sets intersect sum dimensions exceed ambient dimension dim dim range dim dim null dim dim null see dim dim null consequence subspace cone must nontrivial intersection view proposition establish theorem showing implies min high probability condition follows universality result corollary restricted minimum singular value proof appears section similarly establish theorem showing implies min min high probability condition follows specialized argument culminates corollary proof appears section need lavish extra attention case cone subspace indeed dimension differ one invisible final results universality randomized dimension reduction theorem extensions interesting challenge delineate scope universality phenomenon described theorem believe remain many opportunities improving result theorem shows width phase transition thm known width phase transition order min linear map standard normal entries wide phase transition general random linear maps related question whether theorem holds sets whose statistical dimension much smaller ambient dimension empirical evidence location phase transition universal class wider model particular results like theorem may valid structured random linear maps figure suggests probability successful embedding universal conditions observation formalized summary theorem first step toward broader theory universality stochastic geometry niversality estricted inimum ingular value section describes quantitative universality law random linear maps show restricted minimum singular value random linear map takes value every linear map substantial class type result provides information stability randomized dimension reduction restricted minimum singular value problem formulation let frame assumptions fix ambient dimension let nonempty compact subset unit ball let random linear map embedding dimension section goal understand distance random projection origin following definition captures property definition restricted minimum singular value let linear map let nonempty subset restricted minimum singular value respect set quantity inf kat briefly write restricted singular value rsv proposition shows restricted minimum singular value quantity interest studying embedding dimension linear map also productive think measure stability inverting map image particular note restricted singular value generalization ordinary minimum singular value obtain selection excess width functional clear restricted singular value decreases respect set inclusion words restricted singular value depends content set case random linear map following geometric functional provides correct notion content definition excess width let positive number let nonempty bounded subset width quantity inf oymak tropp versions excess width appear early work gordon cor also come recent papers analysis gaussian random linear maps excess width number useful properties results immediate consequences definition subset unit ball width satisfies bounds particular excess width positive negative width weakly increasing implies width decreasing respect set inclusion implies width absolutely homogeneous excess width related gaussian width using also relate excess width statistical dimension term excess width standard formula suggests moniker appropriate according theorem sign excess width indicates random dimension reduction embedding dimension succeeds fails high probability papers develop methods computing excess width variety situations instance subspace sophisticated example consider probability simplex develop asymptotic expression excess width lim inf inf proof formula involved must omit details see framework making computations see part examples also possible estimate excess width numerically approximating expectation empirical average theorem universality restricted minimum singular value preparation present main result universality properties restricted minimum singular value random linear map theorem universality restricted minimum singular value fix parameters model choose parameters number following statement holds suppose ambient dimension nonempty compact subset unit ball embedding dimension range width small random linear map obeys model parameters universality randomized dimension reduction furthermore convex constant depends parameter random matrix model section contains overview argument detailed proof theorem appears section stronger probability bounds hold random linear map drawn model random linear map drawn model theorem asserts distance random projection origin typically much smaller width similarly convex distance random image origin typically much larger conclusions require excess width small compared root embedding dimension theorem iipis vacuous nevertheless case spherically convex condition implies high probability theorem already seen illustration theorem action view expression excess width probability simplex theorem explains experiment documented figure remark prior work random linear map gaussian gordon work cor yields conclusion theorem second conclusion appears recent see korada montanari established universality laws optimization problems arise communications mathematical challenge work similar showing rsv cube universal model adopted arguments work hand cube present technical difficulties arise general sets types random linear maps types sets aware research yields precise value restricted singular value required assert universality literature asymptotic convex geometry mathematical signal processing contains many results restricted singular value provide bounds correct order unspecified multiplicative constants theorem proof strategy one standard template proving universality law factors problem two parts first compare general random structure canonical structure additional properties use extra properties obtain complete analysis canonical structure comparison learn general structure inherits behavior canonical structure recipe describes lindeberg approach central limit theorem see sec recently similar methods directed harder problems universality local spectral statistics random matrix research closest spirit papers let imagine could apply technique prove theorem suppose compact convex set performing standard discretization smoothing steps might invoke lindeberg exchange principle compare restricted singular value random linear map standard normal matrix evaluate restricted minimum singular value standard normal matrix use gaussian minimax theorem cor last incorporate convex duality argument obtain reverse inequality oymak tropp unfortunately simple approach adequate argument ultimately follows related pattern overcome number technical obstacles let explain serious issue mechanism handling would like apply lindeberg exchange principle replace entry standard normal variable problem may contain vector large entries try replace columns associated large components incur intolerably large error moreover given coordinate set may contain vector takes large value distinguished coordinate fact seems foreclose possibility replacing part matrix address challenge dissecting index set small subset define set vectors whose components small first argue restricted singular value subset differ much restricted singular value entire set min may use lindeberg method make comparison matrix hybrid form replace columns listed set independent standard normal variables point need compare minimum singular value restricted subset excess width full set words remaining part original matrix plays negligible role determining restricted singular value perform estimate using gaussian minimax theorem convex duality arguments coarse results nonasymptotic random matrix theory see part iii detailed statements main technical results support theorem proofs results ingredients standard tools modern applied probability combined subtle way make long argument clearer attempted present proof fashion pieces level combined explicitly level theorem extensions expect number avenues extending theorem analysis shows error estimating excess width case standard normal matrix error actually constant order thm improve error bound general random linear maps related question whether thep universality persists excess width small comparison empirical evidence restricted minimum singular value may universality properties class random linear maps wider model proof probably adapted show types functionals exhibit universal behavior example study min convex lipschitz function type optimization problem plays important role statistics machine learning time know many natural functionals exhibit universal behavior particular consider restricted maximum singular value sup kat many sets maximum singular value take universal value linear maps model example see figure interesting challenge determine full scope universality principle uncovered theorem universality randomized dimension reduction part applications part paper outline implications theorems information sciences focus problems involving least squares minimization reduce number distinct calculations need perform nevertheless techniques apply broadly effort make presentation shorter intuitive also chosen sacrifice precision section discusses structured signal recovery particular compressed sensing problem section presents application coding theory section describes results allow analyze randomized algorithms numerical linear algebra finally section gives universal formula prediction error sparse regression problem ecovery tructured ignals andom easurements signal processing dimension reduction arises mechanism signal acquisition idea number measurements need recover structured signal comparable number degrees freedom signal rather ambient dimension given measurements reconstruct signal using nonlinear algorithms take account prior knowledge structure field compressed sensing based idea random linear measurements offer efficient way acquire structured signals practical challenge implementing proposal find technologies perform random sampling universality laws significant implication compressed sensing prove number random measurements required recover structured signal strong dependence distribution random measurements result important applications offer limited flexibility type measurement take theory justifies use broad class measurement ensembles phase transition sparse signal recovery section study phase transition appears reconstruct sparse signal random measurements via minimization large class random measurements prove distribution affect location phase transition result resolves major open question theory compressed sensing let give precise description problem suppose fixed vector precisely nonzero entries let random linear map suppose access image interpret data list samples unknown signal standard approach reconstructing sparse signal data solve convex optimization problem minimize kxk subject minimizing norm promotes sparsity optimization variable constraint ensures virtual measurements consistent observed data say optimization problem succeeds unique solution coincides otherwise fails setting following challenge arises compressed sensing problem optimization problem likely succeed fail recover function sparsity number measurements ambient dimension distribution random measurement matrix question subject inquiry thousands papers last years see books background references series recent papers compressed sensing problem solved completely case random measurement matrix follows standard normal distribution see remark narrative oymak tropp probability exact recovery nonzero rademacher probability exact recovery student dof number measurements theory sparsity theory empirical probability sparsity igure universality recovery phase transition plots depict empirical probability minimization problem recovers vector nonzero entries vector random measurements heatmap indicates empirical probability computed trials white curve phase transition promised proposition eft random measurement matrix sparse rademacher matrix average nonzero entries ight random measurement matrix independent student entries see section details brief exists phase transition function defined inf phase transition function increasing convex satisfies measurement matrix standard normal implies fails probability implies succeeds probability words number measurements increases probability success jumps zero one point range measurements error terms improved substantially presentation suffices purposes donoho tanner performed extensive empirical investigation phase transition minimization problem work suggests gaussian phase transition persists many types random measurements see figure small illustration universality results provide first rigorous explanation phenomenon measurement matrices drawn model proposition universality phase transition assume ambient dimension number measurements satisfy vector exactly nonzero entries random measurement matrix follows model parameters observe vector universality randomized dimension reduction ambient dimension implies fails probability implies succeeds probability suppresses constants depend proposition follows directly universality result theorem established calculation phase transition standard normal setting proof approach quite standard let set descent directions norm point primal optimality condition demonstrates reconstruction succeeds since norm closed convex function set descent directions forms closed convex cone therefore set closed spherically convex follows theorem behavior undergoes phase transition statistical dimension random linear map drawn model result prop contains first complete accurate computation statistical dimension descent cone norm see also prop fact completes proof note proposition requires sparsity level proportional ambient dimension provides information refining argument address case proportional small number depends regularity random linear map empirical work donoho tanner unable provide statistical evidence universality hypothesis regime small remains open problem understand rapidly vanish universality phenomenon fails paper also contains numerical evidence phase transition persists random measurement systems structure model remains intriguing open question understand experiments remark prior work early donoho donoho tanner observed phase transition number standard normal measurements needed reconstruct sparse signal via minimization using methods integral geometry able perform heuristic computation location phase transition function subsequent work proved transition valid parameter regimes later reported extensive empirical evidence distribution random measurement map little effect location phase transition early rudelson vershynin proposed different approach studying minimization adapting results gordon depend gaussian process theory stojnic refined argument obtain empirically sharp success condition standard normal linear maps work establish matching failure condition stojnic calculations clarified extended signal recovery problems papers bayati first paper rigorously demonstrate phase transition valid standard normal measurements argument based state evolution framework iterative algorithm inspired statistical physics work also gives striking conclusion phase transition universal class random measurement maps result requires measurement matrix independent standardized subgaussian entries absolutely continuous respect gaussian distribution consequence paper excludes discrete oymak tropp models furthermore applies minimization contrast proposition proof much wider compass paper amelunxen contains first complete proof standard normal measurements work significant established first time phase transitions ubiquitous signal reconstruction problems also forged first links approach phase transitions based integral geometry based gaussian process theory papers build ideas obtain precise estimates success probability subsequently stojnic refined gaussian process methods obtain detailed information behavior errors noisy variants minimization problem work extended series papers abbasi oymak thrampoulidis hassibi collaborators research detailed understanding behavior convex signal recovery methods gaussian measurements knowledge current paper first work extends type general analysis beyond confines standard normal model signal recovery problems compressed sensing problem prominent example large class related questions universality results implications entire class problems include brief explanation let convex function whose value increases complexity argument norm example complexity measure appropriate sparse signals similarly schatten good complexity measure matrices let vector low draw random linear map suppose access measurements attempt reconstruct solving convex optimization problem minimize subject words find vector minimum complexity consistent observed data say succeeds unique optimal point coincides otherwise fails paper proves phase transition behavior standard normal universality law theorem allows extend result include every random linear map model define set descent directions point ambient dimension implies fails probability implies succeeds probability words phase transition behavior number measurements equals statistical dimension set descent directions point see papers general methods computing statistical dimension descent cone complements conclude section additional remarks scope results signal recovery first discuss geometric applications second mention signal processing problems studied using methods proper convex function takes least one finite value never takes value universality randomized dimension reduction geometric implications proposition understood statement facial structure random projection ball equivalently provides information facial structure convex hull random points suppose ball fix face let random linear map model say preserves face dimensional face projection reinterpret proposition saying implies unlikely preserve implies likely preserve connection minimization facial structure ball identified see also sec another way framing result fix index set cardinality vector entries let independent random vectors drawn model consider absolute convex hull conv question whether set conv face statements implies likely face implies unlikely face see paper discussion connection facial structure polytopes signal recovery universality results type also appear bayati signal processing applications often want perform signal processing tasks data reducing dimension section focused problem reconstructing sparse signal random measurements related problems detection observed signal consist template corrupted noise noise classification observed signal belong class class literature contains many papers propose methods solving problems dimension reduction example see existing analysis either qualitative assumes dimension reduction map gaussian universality laws used study precise behavior compressed detection classification general types random linear maps brevity omit details ecoding tructured rrors one goals coding theory design codes decoding algorithms correct gross errors transmission particular common proportion received symbols corrupted section show large family random codes decoded presence structured errors number errors correct universal family result valuable applies random codebooks closer realistic coding schemes result random decoding also interpreted statement behavior lad method regression also discuss universality results apply class demixing problems decoding sparse errors work random linear code real field consider fixed message let random matrix called codebook context instead transmitting original message transmit coded message suppose receive version message number entries corrupted error vector nonzero entries simplicity assume depend codebook oymak tropp empirical probability decoding message length nonzero rademacher codebook theory number errors empirical probability length code igure universality decoding phase transition plot shows empirical probability method decodes message received message corruption exactly nonzero entries codebook sparse rademacher matrix average nonzero entries heatmap gives empirical probability correct decoding computed trials white curve exact phase transition promised proposition see section details setting one attempt decode message using minimization method solve optimization problem minimize kzk subject words search message sparse corruption match received data say optimization succeeds unique optimal point coincides otherwise fails question optimization problem effective decoding received transmission many errors correct function message length code length following result gives solution problem codebook drawn model proposition universality sparse error correction assume message length code length satisfy message arbitrary error vector exactly nonzero entries random codebook follows model parameters observe vector message length implies implies succeeds probability fails probability function defined suppresses constants depend universality randomized dimension reduction result significant allows understand behavior method sparse discrete codebook type code somewhat closer practical coding mechanism ultrarandom codebooks studied past see remark figure contains illustration theory compares actual performance coding scheme proof analyze decoding problem change variables obtain equivalent optimization problem minimize subject problem decoding procedure succeeds unique optimal point pair introduce set descent directions norm primal optimality condition shows decoding succeeds range let compute statistical dimension first relation polar identity statistical dimension value statistical dimension appears since properties function ensure first demonstrate decoding fails number errors large must show range polarity thm suffices check null probability least relation follows theorem provided finally revert inequality expressed terms last must check decoding succeeds number errors sufficiently small must verify range high probability relation follows ideas closely related proof theorem direct consequence see section details remark regression proposition also interpreted statement performance deviation method fitting models outliers suppose observe data rows interpreted vector measured variables independent subject experiment vector lists coefficients true linear model sparse vector contains small number arbitrary statistical errors deviation method fits model solving minimize proposition shows procedure identifies true model exactly provided number contaminated data points satisfies remark prior work idea using minimization decoding presence sparse errors dates least far back paper scheme received attention work later donoho tanner applied phase transition calculations assess precise performance coding scheme standard normal codebook interpretation result appears sec paper revisits coding problem develops sharp analysis case codebook random orthogonal matrix current paper contains first precise result extends codebooks general distributions oymak tropp demixing problems decoding problem example convex demixing problem universality results used study questions species let proper convex functions measure complexity signal suppose signals low draw random matrices model suppose observe vector interpret random matrices known transformations unknown signals example matrices might denote dictionaries two components sparse attempt reconstruct original signal pair solving minimize max subject words witness superposition two structured signals attempt find lowest complexity pair reproduces observed data demixing problem succeeds unique optimal point equals analyze problem introduce two descent sets scaling descent directions max pair coincide direct product statistical dimension direct product two spherical sets satisfies therefore theorem demonstrates implies fails probability implies succeeds probability words amount information needed extract pair signals superposition equals total complexity two signals result holds true wide class distributions andomized umerical inear lgebra numerical linear algebra nla study computational methods problems linear algebra including solution linear systems spectral calculations matrix approximations last years researchers developed many new algorithms nla exploit randomness perform computations efficiently see surveys overview field section apply universality techniques obtain new results dimension reduction randomized nla discussion shows broad class dimension reduction methods share quantitative behavior therefore within limits choose random linear map computationally appealing design numerical algorithms based dimension reduction added bonus arguments lead new proof limit minimum singular value random matrix drawn model subspace embeddings randomized nla one key primitives subspace embedding subspace embedding nothing randomized linear map annihilate point fixed subspace definition subspace embedding fix natural number let arbitrary subspace say randomized linear map oblivious subspace embedding order high probability term oblivious indicates linear map chosen without knowledge subspace universality randomized dimension reduction definition subspace embedding authors include quantitative bounds stability embedding estimates useful analyzing certain algorithms left essential standard normal matrix provides important theoretical practical example subspace embedding example gaussian subspace embedding natural number standard normal matrix subspace embedding probability one embedding dimension practice preferable select embedding dimension ensure restricted singular value sufficiently positive makes embedding stable see details gaussian subspace embedding superb dimension reduction properties hand standard normal matrices expensive generate store perform arithmetic therefore randomized nla algorithms better use subspace embeddings discrete sparse universality results demonstrate certain range parameters every matrix follows model enjoys subspace embedding properties gaussian matrix proposition universality subspace embedding suppose ambient dimension sufficiently large embedding dimension satisfies random linear map follows model parameters subspace probability particular subspace embedding order whenever expressions suppresses constants depend proof proposition consequence theorem last subspace statistical dimension introduce error term make sure stated result valid hypotheses theorem force note proposition applies sparse rademacher linear map fixed arbitrarily small proportion nonzero entries particular example received extensive attention recent years although works typically focus regime subspace dimension small sparsity level random linear map vanishing proportion embedding dimension remark prior work simple problem considered proposition much sharper results available random matrix literature see paper recent analysis well additional references remark limit minimum singular value one important problems random matrix theory obtain bounds extreme singular values random matrix law gives result case entries random matrix independent standardized reproduce slightly weaker version law minimum singular value modifying proof proposition fix aspect ratio natural number define draw random matrix model fixed parameters apply theorem see oymak tropp denotes kth largest singular value assumptions known empirical distribution singular values converges probability density whose support interval follows therefore may conclude probability comparison law thm gives conclusion almost surely entries four finite moments see recent paper optimal result sketching least squares randomized nla one core applications dimension reduction solve problems perhaps additional constraints idea attributed studied intensively last decade see surveys information section develop sharp bounds simplest version approach suppose fixed matrix full column rank let vector consider problem kax minimize problem expensive solve one remedy apply dimension reduction draw random linear map model consider compressed problem minimize question quality solution depends embedding dimension following result provides optimal estimate proposition randomized least squares error bound instate prevailing notation fix parameters assume number constraints sufficiently large function parameters embedding dimension comparable number constraints embedding dimension somewhat larger number variables high probability solution reduced problem satisfies kax solution original problem particular kax words excess error incurred solving compressed problem negligible compared optimal value problem choose embedding dimension sufficiently large proposition improves substantially recent work cor terms error bound terms assumptions randomized linear map let remark nothing special ordinary least squares also solve problems convex constraint set dimension reduction class problems also obtain optimal bounds adapting argument example see results section universality randomized dimension reduction proof let solution original problem define optimal residual recall orthogonal range moreover minn kax kax without loss generality may scale problem next change variables define note orthogonal write reduced problem minimize close zero dimension reduction effective expect solution since use fact proposition subspace embedding obtain holds high probability number constant depends priori bound nothing need observation ensure optimizing compact set constant radius omit details next invoke theorem bound optimal value reduced problem define compact convex set range let parameter depend parameter high probability min min direct calculation bound excess width let orthogonal projector onto range observe inf inf inf inf first line definition excess width next simplify via orthogonality scaling use translation invariance infimum remove gaussian term apply jensen inequality draw expectation inside infimum note rank make change variables solve scalar convex optimization problem infimum occurs value since may confident shown high probability min prove main result furthermore evidence norm minimizer compute value optimization problem restricted points small positive number chosen later verify optimal value restricted problem usually larger bound optimal value argument implies high probability end define introduce compact set range oymak tropp theorem shows high probability min calculating excess width inf inf add subtract reach second line use fact apply gaussian variance inequality fact bound expectation one infimum occurs objective convex exceeds unconstrained minimizer next simplify expression involving setting find inequality follows linear upper bound square root one combine arrive min comparing last display discover choice sufficient ensure high probability min min follows minimum usually occurs set determine reinterpret inequality obtain stated result considering set vectors range obtain matching lower bound omit details remark prior work idea using random linear maps accelerate solution problems appears work approach extended refined literature randomized nla see surveys overview research concerned randomized linear maps favorable computational properties results much less precise proposition recently pilanci wainright offered refined analysis randomized dimension reduction constrained problems still falls short describing actual behavior methods parts argument adapted work oymak thrampoulidis hassibi universality randomized dimension reduction rediction rror lasso universality results always played important role statistics fundamental example law large numbers justifies use sample average estimate mean general distribution similarly central limit theorem permits build confidence interval mean distribution statistics relies sophisticated methods often based optimization estimate population parameters particular applied statisticians frequently employ lasso estimator perform regression variable selection linear models recently researchers developed theory predict precise behavior lasso data assumed gaussian critical methodological challenge develop universality results expand range models make confident assertions performance lasso section prove first general universality result prediction error using lasso model estimate theory offers justification using lasso model make predictions data derives sparse model expect developments direction play important role applied statistics sparse linear model lasso lasso designed perform simultaneous regression variable selection linear model let present simple statistical model study behavior lasso estimator suppose random variable takes form deterministic vector model parameters nonzero entries random vector predictor variables drawn model noise variance known statistical error drawn model independent interpret family predictor variables want use predict value response variable variables relevant prediction others confounding observed value response contaminated statistical error assumptions idealized let emphasize analysis holds even predictors noise suppose observe independent pairs drawn model let unknown vector statistical errors one goals sparse regression use model coefficients predict future responses data construct estimate given fresh random vector predictor variables predict unknown response using linear estimate want control mean squared error prediction defined msep using statistical model easy verify msep words prediction error controlled squared error estimating model coefficients model coefficients estimator lasso uses convex optimization produce estimate chosen arbitrarily set solutions problem minimize subject oymak tropp mean squared error prediction msep lasso estimate model asymptotic theory rademacher empirical nonzero rademacher empirical student dof empirical empirical msep variance number subjects igure universality lasso prediction error plot shows msep obtained lasso estimator averaged design matrices statistical errors linear model number predictors number active predictors nonzero coefficient model unit magnitude statistical error gaussian variance number subjects varies dashed line marks location phase transition number subjects required identify model exactly noise variance zero gray curve delineates asymptotic upper bound normalized msep proposition markers give empirical estimate trials msep design matrix specified distribution formula rows matrix observed predictor vectors entries vector measured responses simplicity also assume exact side information prove following result squared error lasso estimate sparse coefficient model first universal statement offers precise analysis class statistical models proposition universality lasso prediction error instate prevailing notation choose parameters assume number subjects sufficiently large function parameters number predictors satisfies number subjects satisfies high probability observed data mean squared error prediction satisfies msep function defined furthermore bound sharp proposition gives upper bound msep matches limit obtained gaussian case see figure numerical experiment confirms theoretical predictions proof result appears section assumptions proposition somewhat restrictive number subjects must roughly comparable number predictors condition probably relaxed error bound theorem allow stronger conclusion argument also extended give even precise formulas msep assumptions also note nothing special constraint similar results valid many convex constraints universality randomized dimension reduction proof proposition without loss generality may assume statistical model scaled noise level define sublevel set norm true parameter vector compact convex set contains origin define random matrix note also follows model making change variables rewrite lasso problem form minimize expression depends assumption let optimal point problem leads formula msep referring see formula kuk helpful introduce additional sets parameter define compact typically nonconvex set kuk also define compact convex set kuk observe furthermore implies inclusion holds convex contains origin let constant depends parameters suppose suffices establish inequality prove kuk min min indeed recall minimizer objective let point minimum attained objective convex function decrease line segment muss pass traverses line segment kuk impossible ordering forces objective decrease way fix parameter range select suitable value later theorem demonstrates high probability min second relation holds excess width decreases respect set inclusion observe inf sup last identity holds factor identify gaussian width fix second parameter theorem demonstrates high probability min much calculate excess width oymak tropp last inequality holds fact gaussian width increasing respect set inclusion combine last four displays reach min min follows establish finding parameters end replace gaussian width number depend parameter cone first relation second follows definition last estimate moreover bounds sharp sufficiently close zero therefore need verify choose appropriately adapt analysis proof proposition introduce function proof proposition direct calculation minimized value note close zero case accurate bound furthermore make selection since see bounded constant depends repeating calculation mutatis mutandis combining determine last inequality holds fixed positive constant assumed conclusion confirmed claim value designated follows high probability holds high probability optimizer satisfies kuk therefore formula mean squared error yields bound msep kuk used assumption homogeneity msep must proportional leads stated result finally allow parameters fixed analysis adapted show error bound sharp universality randomized dimension reduction remark prior work special case standard normal design standard normal error result proposition appeared paper extension general random models new nevertheless proof lot common analysis oymak tropp part iii universality restricted minimum singular value proofs theorem theorem part paper present detailed proof universality law restricted singular value random matrix theorem first part universality law embedding dimension theorem section contains main technical result establishes universality bounded random matrix model model section extends result bounded random matrices random matrix model model section obtain theorem immediate consequence result matrices section shows derive theorem additional consequence remaining sections part lay calculations undergird result bounded random matrices estricted ingular values ounded andom atrix key challenges establishing universality restricted minimum singular value already present case random matrix drawn bounded model model section presents universality law bounded random matrices gives overview calculations required establish result theorem main result bounded random matrix model main technical result paper theorem behavior restricted minimum singular value bounded random matrix theorem rsv bounded random matrix model place following assumptions let natural numbers let closed subset unit ball draw random matrix model bound squared restricted singular value following properties squared restricted singular value concentrates mean scale expectation squared restricted singular value bounded terms excess width convex set squared restricted singular value bounded terms excess width furthermore entries need symmetric result hold proof result occupy rest part paper section summarizes required calculations detailed arguments proof theorem concentration theorem states squared restricted singular value concentrates around mean prove claim proposition appears argument depends concentration inequalities derived using entropy method approach less standard move interesting part proof universality randomized dimension reduction setup proof theorem dissection index set let continue proof theorem conclusions overall approach cases details differ first step dissect index set appropriate subsets set introduce closed subset via rule words contains vectors coordinates listed sufficiently small note convexity set implies convexity subset fix number since subset unit ball every vector satisfies therefore belongs subset cardinality decomposition clear number subsets decomposition satisfies must limit cardinality subsets control number subsets need examine proof theorem lower bound rsv let proceed proof theorem lower bound rsv time fix parameter designates cardinality sets require first pass restricted singular value whole set bound depends subsets proposition gives comparison min log used decomposition bound number subsets decomposition invoke proposition main ingredient proof estimate concentration inequality restricted singular values proposition let focus specific subset study restricted singular value want replace entries random matrix standard normal random variables proposition allows make partial progress toward goal let standard normal matrix independent define random matrix log bound requires assumption argument based lindeberg exchange principle used method unusual way vector definition gives control magnitude entries listed exploit replace column submatrix coordinates listed may large replace column submatrix without incurring significant penalty instead leave place next want compare expected rsv side excess width set proposition provides bound second inequality consequence facts excess width decreasing respect set inclusion calculation proposition uses gaussian minimax theorem see oymak tropp fact simplify average respect standard normal matrix also invoke standard results nonasymptotic random matrix theory control expectation obtain simple lower bound last term linearizing convex function point last estimate follows observation inf calculation arbitrary point finally sequence four displays combine error terms obtain log log logarithms optimal choice cardinality parameter since choice ensures conclude log log combine logarithm power adjust constants complete proof theorem proof theorem upper bound rsv convex set high level steps proof theorem similar argument last section many technical details however depend convex duality arguments fix cardinality parameter requirement first step pass min bound trivial consequence facts subset decreasing respect set inclusion select subset invoke proposition exchange entries random matrix standard normal variables using random matrix last section log bound requires assumption proof identical proof lower bound next compare expected rsv side excess width proposition delivers argument complicated analogous lower bound depends convexity also apply assumption proposition allows replace excess width excess width log use decomposition bound number sets invoke proposition proof depends gaussian concentration inequality sequence bounds combine error terms arrive log log universality randomized dimension reduction expand square using reach log log log select arrive log log log combine power logarithm adjust constants finish proof theorem estricted ingular values andom atrix section present extension theorem matrix model model section explain universality result restricted singular value theorem immediate consequence section show derive first half universality result embedding dimension theorem corollary main result random matrix model extend theorem random matrix model model using truncation argument following corollary contains detailed statement result corollary rsv random matrix model fix parameters place following assumptions let natural numbers let closed subset unit ball draw random matrix satisfies model given restricted singular value following properties high probability restricted singular value bounded excess width convex set high probability restricted singular value bounded excess width function strictly positive constant depends proof corollary appears section main idea truncate entry random matrix individually treat bounded part random matrix using theorem show tails negligible means relatively simple norm bound random matrices fact proof theorem corollary theorem easy consequence corollary recall assumptions theorem embedding dimension satisfies closed subset unit ball width satisfies random linear map follows model parameters therefore may apply corollary see convex oymak tropp simplicity dropped embedding dimension side bounds last display prove theorem suffices check make error term smaller select ambient dimension large enough indeed conditions ensure since positive number implies needed show proof theorem corollary theorem also easy consequence corollary recall assumptions theorem compact subset contain origin statistical dimension satisfies random linear map follows model parameters section consider regime embedding dimension need demonstrate null begin reduction specific choice embedding dimension let matrix formed first rows random linear map function null weakly increasing null decreasing sequence sets therefore suffices verify case note statistical dimension introduce spherical retraction proposition demonstrate implies implies therefore check suffices produce lower bound restricted singular value choices corollary yields complete proof need verify hypotheses imply point follows two relatively short calculations since subset unit sphere quickly compute ite excess width inf kxk second identity holds subset unit sphere used relation recall statistical dimension general set defined introduce value embedding dimension meanwhile bound dimensional term universality randomized dimension reduction first inequality holds second inequality uses assumption since positive find number implies combine last display determine claim valid heorem oncentration estricted ingular values section demonstrate restricted minimum singular value bounded random matrix concentrates mean value result yields theorem proposition theorem concentration let closed subset let random matrix drawn model uniform bound proof relies modern concentration inequalities derived using entropy method establish bound lower tail section bound upper tail appears section several parts paper rely variants proposition follow essentially arguments omit details proofs avoid repetition proposition lower tail rsv first establish restricted singular value unlikely much smaller mean proof depends exponential moment inequality derived using massart modified logarithmic sobolev inequality instance see thm thm fact random variable lower tail let independent sequence real random variables nonnegative function define supi sup suppose supi function supp returns support random variable exp fact hand may derive lower tail bound proof proposition eqn introduce random variable min min index pair real number define random matrix replacing entry number random variable supi max max min must evaluate lower variance proxy supi oymak tropp end select point arg index pair supi max min min max matrix differs row therefore expand squared norms components components cancel except one discover supi max max max last inequality holds consequence supi sum pairs arrive sup min reach second line used fact last line depends definition summary demonstrated apply fact need bound expectation designate point calculate identity holds independent standardized entries fact ensures exp exp rewrite formula complete proof proposition upper tail rsv next establish restricted singular value bounded random matrix unlikely much larger mean proof depends another tail inequality random variables result also follows massart modified inequality argument similar proofs thms fact random variable upper tail let independent sequence real random variables function define supi sup suppose supi uniformly fixed value supi universality randomized dimension reduction function supp returns support random variable exp proof proposition eqn proceed proof remains verify uniform bound supi starting calculate supi calculation relies fact entries magnitude bounded apply fact complete argument heorem robability ounds issections section establish results compare expectation minimum random variables indexed set expectations minima indexed subsets facts easy consequences concentration phenomena dissection excess width first show excess width set related excess width collection subsets proposition theorem dissection excess width consider closed subset unit ball decomposed finite number closed subsets holds min min log min words min log proof since stratify minimum min min min obtain lower tail bound minimum combining lower tail bounds minimum integrate tail probability obtain required expectation bound subset contained map min gaussian concentration inequality fact provides tail bound min min apply union bound see min min min log oymak tropp log using latter estimate quickly compute excess width abbreviate min min min log stated result trivial may assume integration parts representation expectation yields reintroduce values combine constants complete argument dissection restricted singular value next show minimum singular value random matrix restricted set controlled minimum singular value restricted subsets proposition theorem dissection rsv consider closed subset unit ball assume decomposed finite number closed subsets let random matrix satisfies model uniform bound min log proof argument similar proof proposition since min min min min lower tail bound restricted singular values implies next take union bound min log log define random variable constant min log stated result trivial may assume calculating proposition simplify bound complete proof universality randomized dimension reduction heorem eplacing ost ntries andom atrix section show possible replace entries random matrix standard normal random variables without changing restricted singular value much proposition theorem partial replacement let random matrix satisfies model magnitude bound let subset cardinality let closed subset implies index define random matrix assuming min min log equivalently min min log usual standard normal matrix hypothesis essential ingredient proof proposition indeed exchange elements random matrix columns indexed depend uniform bound control replacement errors proof proposition proof proposition involves several steps helpful give overview begin earnest throughout discussion index set fixed let discretization parameter first step argument replace index set finite subset cardinality log log obtain bounds min min see lemma details quite standard next introduce smoothing parameter define function log advantageous work differentiable lemma demonstrates pay small cost passing log log argument also standard finally compare expectation evaluated random matrices lemma encapsulates argument based lindeberg principle fact idea replace one entry time corresponding entry gaussian random variable lose little exchange function smooth vectors bounded coordinates oymak tropp combine relations obtain estimate total error min min log used fact log log smoothing term remains identify appropriate values parameters select discretization error negligible choose smoothing parameter summary min min log since third term dominates bound stated proposition discretizing index set first step proof proposition replace set finite subset without changing restricted singular values substantially lemma proposition discretization adopt notation hypotheses proposition fix parameter construct subset cardinality log log property min furthermore bound holds replace proof choose discretization minimal cardinality max since vol vol use classic volumetric argument ensure covering satisfies example see lem replace set covering incur discretization error establish error bound argument identical since immediate min claim min since standardized entries denotes frobenius norm combine last three displays verify quite easy establish claim ksk used standard norm inequalities fact subset unit ball let arg mint let point min universality randomized dimension reduction taking expectations arrive proposition smoothing minimum second step proof proposition pass restricted minimum singular value discrete set smooth function rely exponential smoothing technique common mathematical literature statistical physics lemma proposition smoothing minimum adopt notation hypotheses proposition let set introduced lemma fix parameter define function via log bound also holds replace proof result follows trivial bounds sum definition function summands nonnegative sum exceeds maximum term thus log log kat similarly sum exceed number summands times maximum term log log log kat combine last two displays multiply negative number reach kat log log kat replace random matrix take expectation reach similarly take obtain result proposition exchanging entries random matrix last step proof proposition hardest must demonstrate expectation function change much replace submatrix submatrix lemma proposition exchanging entries adopt notation hypotheses proposition let set described lemma let function parameter random matrix defined proof establish lemma replacing rows random matrix rows random matrix one time let random matrix whose first rows drawn whose remaining rows drawn construction follows demonstrate combining last two displays arrive bound fix index construction random matrices identical except row perform estimate convenient suppress dependence function remaining rows end define functions via log oymak tropp via written rows definitions therefore inequality equivalent since matrix drawn model random vector contains independent standardized entries whose magnitudes bounded meanwhile form matrix shows random vector coincides components indexed entries indexed independent standard normal variables sublemma contains proof inequality lemma comparison principle one row complete argument lemma need control much certain function changes replace entries argument standard normal variables following result contains required calculation sublemma lemma comparison one row adopt notation hypotheses proposition let set defined lemma introduce function given log arbitrary function suppose random vector independent standardized entries bounded magnitude suppose random vector standard normal vector proof sublemma based modern interpretation lindeberg exchange principle similar spirit examples paper apply following version lindeberg principle adapted works fact lindeberg exchange principle let function three continuous derivatives let standardized random variables necessarily independent three finite moments max max proof fact involves nothing taylor expansion around origin terms cancel random variables mean zero variance one inequality hand may proceed proof sublemma proof sublemma without loss generality assume index set indeed entries random matrix independent standardized uniformly bounded matter set columns distinguish choice allows intuitive indexing fixed index define interpolating vector introduce function universality randomized dimension reduction observe first identity holds sum telescopes fact shows max max claim index max max establish bound combine last three displays reach main result follows establish fix index forthcoming sublemma demonstrate expression immaterial note random vector independent precise distribution indeed derivative standard basis vector last inequality holds conditions allow invoke assumption arrive following bound quantity interest max max max merged bounds control next invoked jensen inequality draw expectation maximum last combined expectations reach remains bound expectations side let begin term linear view identity max max used assumption independent factor expectation second term second line bounds third line exploit fact via hypothesis fact also relied estimate indeed jensen inequality allows replace expectation second moment simplifies independent family standardized random variables since norm exceed one oymak tropp continuing term quadratic pursue approach see max max bound relies inequality example see cor remark estimate could improved additional information distribution entries substituting bounds obtain max establishes first branch claim second branch follows similar argument use explicit values moments standard normal variable instead uniform upper bound sublemma derivative calculations final obstacle proof proposition bound derivatives required sublemma argument uses standard methods statistical physics similar approach paper sublemma sublemma derivatives adopt notation hypotheses proposition sublemma let linear function derivative constant vector define function log arbitrary third derivative function satisfies max expression random vector independent proof introduce parameterized probability measure set treat normalizing factor function write derivatives convenient use statistical mechanics notation expectation respect measure function brevity always suppress dependence parameter often suppress dependence well function proportional logarithm normalizing factor log thus straightforward express third derivative terms derivatives universality randomized dimension reduction direct calculation derivative satisfies second derivative third derivative ascertain completes calculation exact form third derivative next simplify formula using basic probability inequalities expectation first consider terms linear observe max indeed jensen inequality allows draw absolute value inside average invoke inequality pull maximum first term similarly since next consider terms quadratic simplest max using jensen inequality find last using lyapunov inequality twice max max introduce last five displays arrive bound max last want replace averages respect averages respect simpler probability measure depend argument relies correlation inequality define another probability measure set oymak tropp write averages respect new measure first consider average let random variable obtained pushing forward measure nonnegative real line since increasing exp decreasing chebyshev association inequality thm provides summary argument shows introduce bound reach inequality statement result follows reinterpret averages respect expectations respect random vector distribution heorem ounding estricted ingular value xcess idth section showed restricted singular value random matrix change much replace hybrid matrix defined contains many standard normal random variables next goal relate restricted singular value hybrid matrix excess width set proposition theorem excess width bound let random matrix model magnitude bound let random matrix independent standard normal entries let subset cardinality let closed subset introduce random matrix min furthermore convex min obtain result invoke gaussian minimax theorem reduce random matrix bounds simpler bounds involving random vectors approach used study ordinary singular values gaussian matrix sec also plays role analysis restricted singular values gaussian matrices application complicated significantly presence matrix universality randomized dimension reduction proof proposition proof proposition involves several steps helpful summarize calculations required first let explain process obtain lower bound general closed subset euclidean unit ball argue min min jensen min lemma min lemma min lemma definition first line pass expected square square expectation reduces technical complexity rest argument next replace gaussian matrix expectation two independent standard normal vectors third remove remaining piece original random matrix arrive formula involving random vector finally replace missing coordinates random vector obtain bound terms excess width arrive inequality upper bound follows similar calculation assuming convex may calculate min min lemma min lemma min lemma min lemma definition first step type inequality requires specialized concentration results restricted singular value hybrid matrix second step chain involves convex duality argument present proof lower bound used assumption simplify error term pass second line third altogether yields bound proposition moment comparison inequality hybrid rsv require moment comparison inequality pass expected square restricted singular value hybrid matrix expectation restricted singular value lemma proposition moment comparison adopt notation hypotheses proposition min min proof define random variable min oymak tropp need bound obtain result applying moment comparison inequality conditional apply moment comparison inequality sublemma applied conditionally choice gives function gaussian variance inequality fact implies combine last two displays adjust constant complete proof proof lemma requires separate moment comparison result random variable depends original matrix sublemma lemma moment comparison original matrix adopt notation hypotheses lemma fixed matrix min min proof define random variable min first observe last inequality jensen use concentration bound quantity mutatis mutandis repeat proof equation proposition see invoke subadditivity square root change variables jensen inequality integration parts deliver third inequality follows introduce last display reach rearrange relation complete proof proposition role gaussian minimax theorem prove proposition must replace gaussian matrix quantity interest pair gaussian vectors key argument following technical result lemma proposition application gaussian minimax theorem adopt notation hypotheses proposition let fixed matrix let independent standard normal random vectors min min furthermore convex min min side trivially equal one universality randomized dimension reduction result depends gaussian minimax theorem see fact statement lemma similar early results gordon cor detailed argument adapted thm see also stojnic proof basic idea express quantity interest value problem tjc min min max apply gaussian minimax theorem obtain probabilistic lower bounds convex also invoke convex duality interchange minimum maximum leads complementary bounds proceed approach however convenient work slightly different minimax problem define deterministic function let standard normal random variable independent everything else introduce two centered gaussian processes kuk kuk indexed let verify processes satisfy conditions required gaussian minimax theorem fact first parameters second parameters kuk inequality formulas verify conditions gaussian minimax theorem fact delivers bound min max min max estimate involve small technicality apply gaussian minimax theorem finite subset must make approximation argument pass entire set omit details let determine values problems first min max min max kuk min similarly min max min oymak tropp next remove term involving side condition calculate min max min max min min min hand min max min introduce last two displays reach min min take complements probabilities rearrange conclude correct second result requires additional duality argument replace function random processes negations variance calculations remain valid particular relations permit apply fact roles reversed step yields minm max minm max let examine problems since bilinear sets compact convex sion minimax theorem allows interchange minimum maximum thus minm max max minm min max min similarly minm max min combining last three displays reach min min proceeding conditioning sign may remove dependence side obtain min min clearly inequality useful case right hand side equal min confirm true universality randomized dimension reduction proposition reducing gaussian matrix gaussian vectors next goal convert probability bounds lemma expectation bounds lemma gives lower bound valid every subset unit ball lemma gives upper bound valid also convex lemma proposition reducing gaussian bound adopt notation hypotheses proposition let fixed matrix let independent standard normal vectors min min particular min min proof make abbreviations min min first note last inequality jensen bound side use integration parts formula lemma finally make estimates var last inequality follows gaussian variance inequality fact indeed function set contained unit ball summary arrive advertised bound lemma proposition reducing gaussian bound adopt notation hypotheses proposition let fixed matrix let independent standard normal vectors min min particular min min proof follow pattern lemma using notation calculate case invoke lemma obtain penultimate inequality oymak tropp proposition removing remaining part original random matrix next step proof proposition remove remaining section random matrix bounds lemmas lemma proposition removing original matrix adopt notation hypotheses proposition let random matrix independent standardized entries satisfy assumption min min particular conclusion valid proof abbreviate observe therefore deterministic bounds random matrix independent standardized entries satisfy fact shows extreme singular values satisfy probability bounds inequalities allow treat singular values equal may perform following estimates min min min max min first inequality add subtract maximum singular value last recall subset unit ball set calculate split integral change variables second integral first integral use trivial bound one probability invoke probability inequality combining last two displays collecting constants arrive min min entirely similar argument delivers matching lower bound together estimates complete proof universality randomized dimension reduction proposition replacing coordinates missing excess width last step proof proposition examine functional lemma involves term must show term change much reintroduce coordinates listed lemma gives easy proof lower bound lemma contains upper bound lemma proposition missing bound adopt notation hypotheses proposition min min proof result immediate consequence jensen inequality min min min min rely fact independent standard normal vectors lemma proposition missing bound adopt notation hypotheses proposition min min proof calculation also easy min min min min last step used fact roof orollary heorem runcation section show establish corollary consequence theorem proof depends truncation argument proof corollary fix parameters assume matrix follows model parameters let compact subset unit ball prove lower bound minimum singular value restricted convex entirely similar approach yields corresponding upper bound fix truncation parameter satisfies decompose random matrix applying truncation described lemma separately entry procedure ensures contains independent symmetric standardized entries bounded words follows model tail contains independent centered entries variance bounded whose pth moment bounded control restricted singular values using triangle inequality min min bound restricted singular value bounded matrix using theorem bound apply simple norm estimate fact based matrix rosenthal inequality thm oymak tropp since follows model theorem give probability bound select make tail probability negligible taking square roots inside event reach step depends subadditivity square root meanwhile entries centered variances pth moments bounded therefore apply norm bound random matrices fact see log log define positive quantity via relation select obtain log log key point arrange order high probability combine reach log log set truncation parameter equate exponents two terms depend simplify using obtain log note powers event bounded away absorb logarithm increasing power slightly furthermore introduce function strictly positive reach inequality constant depends parameter exponential vanishes faster polynomial combine terms side complete proof upper bound use theorem control expectation restricted singular value case error term expectation bound becomes change presents new difficulties arrive result slight modification argument corollary truncation individual random variables section describe truncation procedure scalar random variables arguments entirely standard include details completeness universality randomized dimension reduction lemma corollary truncation let random variable satisfies properties listed model let parameter satisfies decomposition proof define random variable since standardized symmetric also standardized symmetric ensure decomposition holds must set random variable also centered symmetry establish properties need calculate expectations first using integration parts markov inequality last step used assumption similar calculation shows last relation holds standardized follows last estimate holds assumption prepared verify uniform bound last inequality follows next need bound variance first identity holds two indicators orthogonal random variables second relation uses expectation calculation estimate dropped indicator second expectation last estimate holds standardized oymak tropp last need check moment inequality estimate follows applying triangle inequality definition dropped indicators invoking triangle inequality finally introduced estimate universality randomized dimension reduction part universality embedding dimension proof theorem part paper develops condition random projection set fails high probability argument establishes second part universality law embedding dimension theorem section contains main technical result condition bounded random matrix maps point set origin section extends argument random matrix model model section use latter result derive theorem remaining parts section lay supporting argument hen mbedding fails ounded andom atrix section introduce functional whose value determines whether linear transformation maps point set origin present main technical result gives estimate functional evaluated random linear map model rest section outlines main steps proof result rap functional dual condition failure study linear map maps point set origin use approach based polarity let make following definition definition range avoids polar rap let closed convex cone let matrix define quantity min note range inner minimum involves polar cone refer rap functional see rap functional important consider closed spherically convex subset cone subspace second conclusion proposition states cone implies third conclusion proposition gives similar result case subspace therefore obtain sufficient condition maps point zero providing lower bound rap functional theorem main result bounded random matrix model main technical result part paper theorem behavior rap functional bounded random matrix theorem rap bounded random matrix model place following assumptions let natural numbers min let closed convex cone define draw random matrix model bound squared rap functional following properties squared rap functional deviates mean scale expected square rap functional bounded log result use symmetry assumption model proof theorem long even though borrow lot proof theorem section contains overview calculations required forward references detailed arguments oymak tropp proof theorem lower tail bound theorem states quantity deviate substantially mean proof similar proof bound proposition shows squared restricted singular value deviate substantially mean omit repetitive details proof theorem truncation dissection let proceed proof theorem define sets radius crad universal constant crad next construct family closed convex subsets define fix cardinality parameter min proof theorem decompositions furthermore maintain heuristic cardinality much smaller either ambient dimension proof theorem lower bound rap functional bound quantity must combine estimates several technical results give outline calculation details postponed series propositions first must account error incur truncate cone wedge proposition demonstrates min min min inequality based estimate norm point inner minimum achieved well probability bound norm random matrix next pass minimum full sets minima subsets min min min min min log proof inequality hews argument proposition need invoke concentration inequality theorem lieu concentration inequality proposition take account bound number subsets decomposition details omitted prepared perform exchange argument pass matrix matrix contains many standard normal entries fix subsets cardinality introduce random matrix standard normal matrix proposition gives bound min min min min log log min proof similar proof proposition discretize sets smooth minima using function apply lindeberg principle main distinction universality randomized dimension reduction replace even less matrix second line immediate consequence facts continue must identify geometric functional hiding within expression write proposition demonstrates min log log proposition main tool gaussian minimax theorem fact allows break standard normal matrix simpler quantities proof requires convex duality arguments well delicate considerations arise last linearize function log log log employed observation max standard normal vector sequence estimates arrive log log log select results bound log log note min required assumed min obtain result quoted theorem hen mbedding fails andom atrix section extend theorem matrix model model section show derive second half universality result embedding dimension theorem consequence corollary main result random matrix model following corollary extends theorem include random matrices drawn model corollary rap random matrix model fix parameters place following assumptions let natural numbers min let closed convex cone define draw random matrix satisfies model given rap functional satisfies probability bound function strictly positive strictly positive constants depend proof corollary follows theorem kind truncation argument appears section omit details oymak tropp proof theorem corollary theorem easy consequence corollary let restate assumptions theorem compact subset contain origin statistical dimension satisfies random linear map follows model parameters must consider regime embedding dimension need demonstrate null section probability decreasing function embedding dimension may well consider case easy see particular min introduce spherical retraction cone subspace replace subset property cone subspace one dimension fewer cone proceed argument proposition relation show cone implies implies therefore verify enough produce lower bound rap functional cone choices cone corollary yields cone need check lower bound rap functional positive point follows two relatively short calculations since subset unit sphere quickly compute excess width using justifications section easily bound dimensional term first inequality holds last two relations rely assumption since positive find number implies combine last result see claim holds true proof proposition application rap functional decoding structured errors section establishes success condition proposition precisely show implies range probability set derived descent cone norm calculation shows begin assume parameters statistical dimension message length universality randomized dimension reduction section conditions imply null probability least indeed statement corresponds change variables polarity conclusion holds remains show parameter choice implies two assumptions made proof proposition condition ensures similarly condition holds equivalently since choose arbitrarily small constant suffices observation completes argument heorem runcation one section argue functional change much truncate cone replacing unbounded set compact set allows develop discretization arguments proposition rap truncation cone adopt notation hypotheses theorem let crad min min min proof since easy see meanwhile triangle inequality gives bound ksk ksk follows arg min implies since norm random matrix concentrates bound shows norm minimizer unlikely large positive parameter observation allows calculate min min min min min min max max reach last line write indicator function terms complement oymak tropp bound second term side crude estimates suffice max max last inequality since satisfies condition fact implies particular using integration parts furthermore constant crad crad set crad max max introduce estimate complete argument heorem eplacing ost ntries andom atrix section show replace entries random matrix standard normal variables without changing value functional substantially proposition rap partial replacement let random matrix satisfies model magnitude bound fix parameter crad let subset cardinality let closed subset implies index let subset cardinality let closed subset implies index suppose matrix block form log min min min min usual standard normal matrix proof proposition fix sets proof proposition error three components min min min min lemma log lemma lemma universality randomized dimension reduction first error comes discretizing sets level second error appears replace minima function parameter last error emerges lindeberg exchange argument complete proof set make discretization error negligible select smoothing parameter arrive min min min min log since crad second term dominates reach stated result proposition discretizing index sets first step proof proposition replace index sets finite subsets lemma proposition discretization adopt notation hypotheses proposition fix parameter contains finite subset contains finite subset whose cardinalities satisfy log log log furthermore subsets property min min bound also holds replace proof choose since subset subset sure coverings cardinality see lem rest proof essentially lemma omit details proposition smoothing minimum next step proof proposition pass minimum function lemma proposition smoothing adopt notation hypotheses proposition let sets introduced lemma fix parameter introduce function log log log estimate also holds replace proof proof almost identical lemma proposition exchanging entries random matrix main challenge proof proposition exchange entries random matrix entries lemma proposition exchange adopt notation hypotheses proposition let function defined lemma oymak tropp proof similar lemma time replace rows listed incur error rows suffices control error exchanging single row following sublemma achieves goal sublemma lemma comparison one row adopt notation hypotheses proposition let sets defined lemma introduce function given log arbitrary function suppose random vector independent standardized entries bounded magnitude suppose random vector standard normal vector proof result much proof sublemma two points require care first use slightly different result compute derivatives sublemma lemma derivatives adopt notation hypotheses proposition sublemma let linear function derivative constant vector define function log arbitrary third derivative function satisfies max random vector depend second making bounds need exploit control magnitude coordinates note second inequality holds similarly crk repeating arguments sublemma obtain bounds form max first two terms dominate third choice crad implies arrive statement sublemma heorem ounding rap unctional xcess idth difficult part proving theorem identify excess width replace original matrix hybrid matrix defined following result job universality randomized dimension reduction proposition theorem excess width bound adopt notation hypotheses proposition let min log log random matrix defined proof proposition occupies rest section highest level proof similar argument underlying proposition write quantity interest minimax apply gaussian minimax theorem replace gaussian matrix pair gaussian vectors afterward analyze resulting expression identify gaussian width new challenges appear step proof proposition overview calculations perform detailed justifications appear upcoming subsections let abbreviate chain inequalities min min max min max min max log log lemma lemma lemma lemma standard convex duality argument next line follows write quantity interest explicitly reach fourth line apply gaussian minimax theorem usual way replace random matrix two standard normal vectors rough sense remaining part random matrix negligible term generates gaussian width defined term contributes dimensional factor apply increasing convex function inequality last display invoke jensen inequality draw expectation notice max finally point completes proof proposition duality rap functional first step argument apply minimax inequality pass formulation amenable analysis gaussian minimax theorem lemma proposition duality adopt notation hypotheses proposition point max proof write norm maximum max oymak tropp minimax inequality allows interchange maximum minimum max min max max value equals zero otherwise takes value step uses assumption closed convex conclude max stated result proposition reducing gaussian matrix gaussian vectors argument similar proof lemma replace gaussian block two gaussian vectors lemma proposition reducing gaussian matrix adopt notation hypotheses proposition min max min max independent standard normal vectors proof new ideas bound refer reader lemmas pattern argument proposition finding gaussian width prove proposition difficulty arises seek good lower bound minimax problem appears lemma following result lemma proposition finding gaussian width adopt notation hypotheses proposition define set min max log log proof proof bound lengthy break argument several steps overall result follows sequence inequalities sublemmas consolidate error terms lemma simplifying minimax first step proof lemma simplify minimax identify key terms sublemma lemma simplifying minimax adopt notation hypotheses lemma min max min max proof let introduce notation quantity interest min max introduce operator fact ensures minimax nonnegative universality randomized dimension reduction first step argument reintroduce missing piece random vector adding subtracting quantity inside operator obtain bound max min max min max second inequality holds subset unit ball step similar proof lemma next combine terms involving row vector since kuk tjc hjc hjc random matrix side independent standardized entries satisfy subgaussian estimate bound repeating calculations see hjc apply estimate inside minimax use arrive lower bound min max hjc min max second line simply consolidated terms lemma simplifying minimax next step proof lemma reduce minimax problem sublemma scalar optimization problem sublemma lemma simplifying minimax adopt notation hypotheses lemma min max min max max min max proof introduce notation max develop two lower bounds maximum coupling random matrix different ways afterward combine results single bound first place choose row vector depends remaining gaussian vector arg max oymak tropp since obtain bound max max second term side satisfies indeed random vector stochastically independent random matrix independent standardized entries second bound even simpler since max min max expression variable introducing arrive min max max min max min max max min max zero branch maximum accounts operator second line follows also introduced new parameter stand lemma probabilistic bounds last major step lemma develop probabilistic bounds terms arise sublemma sublemma lemma probabilistic bounds adopt notation hypotheses lemma min max max min max min max log log proof introduce notation min max max min max assume permitted final result would otherwise become vacuous next stage argument introduce probabilistic bounds branches maximum use control expectation convenient abbreviate max min max note since gaussian concentration inequality fact implies hand sublemma demonstrate log log universality randomized dimension reduction therefore taking complements union bound log log nonnegative random variable number holds using estimate probability bound find log min max log min max log log used maximum combine error terms convenience also dropped zero branch maximum introduced number two numerator second branch lemma solving scalar minimax problem final step proof lemma solve scalar minimax problem emerges sublemma sublemma lemma solving minimax problem adopt notation hypotheses lemma min max log log proof first branch maximum increasing second branch decreasing minimum occurs two branches equal provided situation occurs setting branches equal identify point saddle value achieved log quickly verify minimax takes value log log required estimate sublemma probabilistic lower bound bilinear minimax section explain obtain lower bound minimax problem terms gaussian width set sublemma sublemma probability bound bilinear minimax assume let random matrix satisfies model bound let standard normal let subset unit ball min max log log proof fix let unit sphere cardinality net satisfies log log standard volumetric argument lem estimate quantity interest discretizing parameter since subsets unit ball bound min max min max establish probabilistic lower bound side oymak tropp first develop probability bound second term side simple spectral norm estimate suffices random matrix standardized entries standard normal usual denotes frobenius norm markov inequality implies follows used facts let turn second quantity side develop strong probability bound fixed point extend entire net using union bound technical reasons easier treat random separately random vector fix point decompose construct random vector satisfies arg max may calculate max max last estimate relies fact ksk second term side easy handle using gaussian concentration inequality fact indeed function mean zero random vector stochastically independent norm bounded one interpret first term side empirical takes work compare quantity gaussian width sublemma contains bound expectation sublemma contains tail bound together deliver probability inequality max log words empirical width comparable gaussian width modulo logarithmic factor introduce two probability bounds deterministic estimate arrive max log finally take union bound pobtain estimate uniform net recall log log select log reach min max log log numerical estimate holds log universality randomized dimension reduction two probability bounds hold simultaneously probability least therefore substitute results adjust constants obtain stated bound sublemma lower estimate empirical width next sublemma demonstrates gaussian width set logarithmic factor larger empirical width set computed bounded random variables step argument bounded random matrices requires symmetry assumption sublemma sublemma empirical width bound adopt notation hypotheses sublemma let fixed vector max log proof sublemma define random vector goal compare empirical width set computed using vector gaussian width set first develop lower bound first moment entries fix index since entries independent symmetric independent family rademacher random variables independent khintchine inequality allows compare first moment second moment since unit norm regard sum weighted average invoke jensen inequality draw average square root last note entries standardized bounded thus let bound functional since independent symmetric coordinates max max max max independent family rademacher random variables independent using corollary contraction principle lem max mini max max applying contraction principle eqn randomize sum independent gaussian variables max max log standard normal vector identify gaussian width complete proof oymak tropp sublemma lower tail empirical width last present concentration inequality empirical width sublemma sublemma lower tail empirical width adopt notation hypotheses sublemma let fixed vector max max direct proof result relies version talagrand inequality thm obtained transportation cost method fact talagrand inequality let metric space suppose fulfills lipschitz bound auxiliary functions satisfy let independent sequence random variables taking values define proof sublemma let interval equipped euclidean distance matrix introduce function max select point arg fact entries bounded magnitude choice see belong euclidean unit ball invoke fact complete argument universality randomized dimension reduction part back matter two appendices contain statements results use throughout paper appendix presents facts gaussian analysis appendix describes spectral bounds random matrices independent entries conclude acknowledgments list works cited ppendix ools aussian nalysis make extensive use methods gaussian analysis provide precise information behavior functions gaussian random variables results come many places paper collected concentration gaussian lipschitz functions begin two concentration results apply lipschitz function independent gaussian variables recall function lipschitz constant also say briefly first result thm gives bound variance lipschitz function second result thm provides normal concentration inequality lipschitz functions fact gaussian variance inequality suppose lipschitz constant let standard normal random vector var equivalently fact gaussian concentration inequality suppose lipschitz constant let standard normal random vector gaussian minimax theorem compute expectations certain functions gaussian random variables depend comparison principle due gordon thm let abstract set family real random variables called centered gaussian process element mean zero finite subcollection jointly gaussian distribution fact gaussian minimax theorem let finite sets consider two centered gaussian processes indexed choices indices suppose real numbers min max min max fact extends infinite index sets approximation oymak tropp ppendix pectral ounds andom atrices argument also depends heavily bounds spectrum random matrix independent entries results give rough estimates adequate purposes first result gives tail bounds extreme singular values rectangular matrix independent subgaussian entries fact subgaussian matrix tail bounds let random matrix independent standardized entries uniformly subgaussian sup largest singular value largest singular value satisfy bounds result follows thm track role subgaussian constant proof second result gives tail bound norm matrix independent entries may two moments based matrix rosenthal inequality thm standard concentration inequality thm fact matrix norm bound fix parameter log let random matrix independent entries following properties entries centered variances entries uniformly bounded var entries uniformly bounded pth moments log log proof sketch write random matrix sum independent random matrices matrix one position zeros elsewhere straightforward application matrix rosenthal inequality thm yields log maxi log log log log log second line follows replace maximum sum exploit uniform moment estimate third line inequality geometric arithmetic means standard concentration inequality moments thm gives expression variance parameter universality randomized dimension reduction entry independent copy corresponding entry remaining entries two matrices applying usual method see maxi applying considerations last paragraph obtain combine results apply markov inequality obtain tail bound introduce estimate expected norm complete argument cknowledgments authors would like thank david donoho surya ganguli babak hassibi michael mccoy andreas maurer andrea montanari ivan nourdin giovanni peccati adrian jared tanner christos thrampoulidis madeleine udell helpful conversations also thank anonymous reviewers editors careful reading suggestions generously supported simons institute theory computing nsf award jat gratefully acknowledges support onr award gordon betty moore foundation eferences amelunxen lotz mccoy tropp living edge phase transitions convex programs random data inf inference bourgain dirksen nelson toward unified theory sparse dimensionality reduction euclidean space geom funct brazitikos giannopoulos valettas vritsiou geometry isotropic convex bodies volume mathematical surveys monographs american mathematical society providence boucheron lugosi massart concentration inequalities oxford university press oxford nonasymptotic theory independence foreword michel ledoux bayati lelarge montanari universality polytope phase transitions message passing algorithms ann appl bayati montanari lasso risk gaussian matrices ieee trans inform theory bogachev gaussian measures volume mathematical surveys monographs american mathematical society providence bai silverstein spectral analysis large dimensional random matrices springer series statistics springer new york second edition bai yin limit smallest eigenvalue sample covariance matrix ann chen donoho saunders atomic decomposition basis pursuit siam sci chatterjee generalization lindeberg principle ann chandrasekaran recht parrilo willsky convex geometry linear inverse problems found comput romberg tao robust uncertainty principles exact signal reconstruction highly incomplete frequency information ieee trans inform theory rudelson tao vershynin error correction via linear programming foundations computer science focs annual ieee symposium pages oct chandrasekaran sanghavi parrilo willsky incoherence matrix decomposition siam clarkson woodruff low rank approximation regression input sparsity time stoc proceedings acm symposium theory computing pages acm new york davenport duarte wakin laska takhar kelly baraniuk smashed filter compressive classification target recognition proc spie volume pages oymak tropp donoho gavish montanari phase transition matrix recovery gaussian measurements matches minimax mse matrix denoising proc natl acad sci usa donoho huo uncertainty principles ideal atomic decomposition ieee trans inform theory donoho johnstone montanari accurate prediction phase transitions compressed sensing via connection minimax denoising ieee trans inform theory donoho compressed sensing ieee trans inform theory donoho large underdetermined systems linear equations minimal solution also sparsest solution comm pure appl donoho centrally symmetric polytopes neighborliness proportional dimension discrete comput davidson szarek local operator theory random matrices banach spaces handbook geometry banach spaces vol pages amsterdam donoho tanner thresholds recovery sparse solutions via minimization information sciences systems annual conference pages march donoho tanner observed universality phase transitions geometry implications modern data analysis signal processing philos trans soc lond ser math phys eng electronic supplementary materials available online donoho tanner counting faces randomly projected polytopes projection radically lowers dimension amer math donoho 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thrampoulidis hassibi generalized lasso precise analysis communication control computing allerton annual allerton conference pages oct available http pilanci wainwright randomized sketches convex programs sharp guarantees information theory ieee transactions sept rotar certain limit theorems polynomials degree two teor verojatnost rudelson vershynin sparse reconstruction fourier gaussian measurements comm pure appl integral geometry spaces constant curvature repub argentina publ comision nac energia atomica ser integral geometry geometric probability publishing reading foreword mark kac encyclopedia mathematics applications vol sarlos improved approximation algorithms large matrices via random projections foundations computer science focs annual ieee symposium pages oct gesammelte mathematische abhandlungen band verlag basel sion general minimax theorems pacific stojnic various thresholds compressed sensing available http july stojnic regularly random duality available http mar 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tracking rigid body internal rotors mar maithripala orientation trajectory tracking rigid body external actuation well studied literature geometric setting well tracking control law relies fact rigid body simple mechanical system sms group special orthogonal matrices however problem designing feedback control laws tracking using internal actuation mechanisms like rotors control moment gyros received lesser attention geometric point view internally actuated rigid body simple mechanical system evolves level set momentum map note propose novel proportional integral derivative pid control law rigid body internal rotors achieves tracking feasible trajectories almost initial conditions ntroduction spacecrafts actuated either internal external mechanisms external mechanisms include gas jet thrusters internal mechanisms include spinning rotors control moment gyros recent past increased interest design control laws simple mechanical systems defined section evolve lie groups results stabilization rigid body sms desired configuration using proportional plus derivative control found geometric tracking specific mechanical systems quadrotor sms rigid body found tracking reference trajectory sms lie group implies tracking reference almost initial conditions tangent bundle lie group general result asymptotic tracking agat sms class compact lie groups found results rigid body assumed externally actuated control torque supplied actuators gas jets stabilization tracking externally actuated rigid body therefore sufficiently well studied problem however problem geometric tracking internally actuated rigid body received much less attention interconnected mechanical systems studied context spherical mobile robots stabilization internally actuated rigid body studied shown feedback torque nayak systems control engineering indian institute technology bombay india aradhana banavar currently visiting professor department electrical engineering indian institute technology gandhinagar maithripala university peradeniya sri lanka smaithri externally actuated rigid body realised internal rotors attached rigid body paper illustrates fact despite feedback forces acting externally actuated rigid body still hamiltonian behaves like heavy rigid body words certain class feedback torque applied rotors rigid body rotors fully actuated sms motivates study agat problem rigid body rotors agat sms studied extensively asymptotic stabilization agas problem rigid body internally mounted rotors solved using proportional plus derivative control rigid body tracking achieved using external internal actuation using local representation rotation matrices trajectory tracking problem considered hoop robot internal actuation pendulum shown class internal actuation configurations exist underactuated mechanical system converted fully actuated sms feedback torques agat sms lie group often achieved proportional plus derivative type control configuration error chosen lie group help group operation along compatible navigation function navigation function morse function unique minimum closed loop error dynamics control force proportional negative gradient covector field generated navigation function plus dissipative covector field sms control drives error dynamics lifted minimum navigation function tangent bundle lie group lifted saddle points maxima navigation function critical points morse function isolated convergence almost global compatibility conditions ensure error function symmetric achieves minimum two configurations coincide authors propose integral action existing control law addition integral term makes control law robust bounded parametric uncertainty rigid body internal rotors underactuated interconnected simple mechanical system control torque provided rotors gets reflected interconnection mechanism rigid body due absence external forces total angular momentum conserved restriction implies certain class angular velocities attained configuration also due presence quadratic rotor velocity terms rigid body alone sms isolate rigid body dynamics introduction feedback control terms system dynamics closed loop rigid body dynamics fully actuated sms thereafter apply existing agat control rigid body obtain corresponding rotor trajectories rotor dynamics control objective track suitable reference trajectory rotor speeds allowed arbitrary paper organised section presenting mathematical preliminaries derive equations rigid body external actuation internal rotors section iii append integral term control law agat sms lie group propose agat control rigid body rotors admissible class reference trajectories section present simulation results proposed control law reliminaries section introduces conventional mathematical notions describe simple mechanical systems found riemannian manifold denoted smooth connected manifold metric denotes riemannian connection flat map given sharp map dual given therefore basis gij gij definition sms simple mechanical system sms smooth manifold metric denoted potential function external uncontrolled force collection covector fields control set system fully actuated span governing equations sms without control input given gradv system trajectory let lie group let denote lie algebra let left group action first argument defined infinitesimal generator corresponding exp defined exp denotes exponential map lie bracket adjoint map defined let isomorphism lie algebra dual inverse denoted induces left invariant metric denote define following equations motion sms derived given describes system trajectory called body velocity dynamics rigid body external actuation consider rigid body external actuation provided gas jets mounted principal axes let denote moment inertia rigid body body frame control vector field applied gas jets system trajectory body velocity sms given equations motion given follows uext dynamics rigid body internal actuation rigid body rotors mounted principal axes rigid body fig rigid body rotors internal actuation case rigid body rotors configuration space configuration variable denoted rotors assumed mounted principal axes rigid body shown figure moment inertia rigid body rigid body frame diag inertia matrix rotors rigid body frame moment inertia ith rotor absence potential energy lagrangian chosen kinetic energy given manifold trivial principal bundle lie group shape space using trivialization shown locally diffeomorphic therefore diffeomorphic lagrangian invariant action reduces function function local trivialization coordinates coordinates coordinate therefore reduced lagrangian variational principle reduced lagrangian applied dividing variations get usual euler lagrange equations obtain euler poincare equations equations motion together called hamel equations uint uint control input applied rotors reconstruction equation given substituting uint body angular momentum reconstruction equation attitude spacecraft angular displacement rotors given uint therefore underactuated sms iii eometric objects admissible trajectories mechanical connection expressed terms connection coefficient body frame obtained details therefore locked inertia tensor body frame obtained inertial frame observed defined details result found therefore momentum map rigid body frame momentum map inertial frame gives conserved angular momentum assume reference trajectory generated another rigid body rotors spatial angular momentum system reference trajectory must lie level set words set reference trajectories given respectively remark written lagrangian invariant respect action defined fig relationship mechanical connection locked inertia tensor momentum map principle fibre bundle momentum map gives conserved quantity along trajectories system given trd agat control section first state result agat fully actuated rigid body external actuators subsequently extend agat rigid body internal rotors definition function lie group navigation function unique minimum critical points det whenever definition configuration error lie group map defined lgr definition consider lie group navigation function configuration error map compatible navigation function tracking problem symmetric minimum navigation function identity agat problem rigid body rotors solved two parts first part pid control law proposed agat rigid body external actuation second part feedback control law chosen equations rigid body rotors reduce externally actuated rigid body first problem well addressed literature proportional derivative control law achieves agat reference trajectory introduce additional integral control term tracking control along lines following theorem theorem agat sms lie group let compact lie group isomorphism lie algebra consider sms riemannian manifold given smooth reference trajectory bounded velocity lie group let navigation function compatible error map exists open dense set agat achieved given following equation constants chosen shown appendix defined proof appendix theorem agat rigid body rotors consider rigid body rotors smooth bounded reference trajectory given let navigation function compatible error map exists open dense set agat achieved uint given following equation uint uext skew positive definite symmetric matrix defined proof substituting second equation first therefore uint uint left hand side sms similar body velocity rigid body right hand side control field designed agat rigid body therefore theorem applicable objective agat rigid body rotors set uext uext obtained choosing trace positive definite symmetric matrix configuration error defined shown navigation function compatible details result found rotor dynamics given substituting remark externally actuated rigid body allowed bounded parametric uncertainty inertia actuation models agat achieved proposed pid control law rigid body rotors however presence bounded parametric uncertainty bounded constant disturbances leads convergence shown interconnected mechanical systems imulation results consider rigid body rotors following parameters diag following initial conditions reference trajectory generated dummy rigid body rotors following parameters diag following initial conditions given momentum conservation equation constant input vector field uint applied dynamics dummy rigid body given order find use bounds appendix max choose find subsequently simulation results shown figure https video link showing tracking axis representation quaternion representation cases reference trajectory red controlled trajectory blue plots show trajectories matrix representation reference trajectory generated dummy rigid body uint sin cos sin fig representative plots uint sin cos sin input torque reference rigid body reference existing refrence existing reference existing uint simulations shown figures respectively compare control effort considering norm uext agat control rigid body trace function considered navigation function values simulations trajectories tracking reference plotted figure control law internal actuation obtained agat tracking law compared proposed control figure reference existing reference existing fig comparison tracking results uint input torque reference rigid body trajectories ries fig representative plots uint torque reference rigid body defined therefore consider ecl given ecl input existing reference trajectories ries fig representative plots uint input torque reference rigid body ppendix define error trajectory defined error dynamics sms constants determined shortly following shows ecl negative essentially follow proof outlined remark objective show ecl along trajectories derivative first term given therefore appropriately weighted cross terms must introduced ecl negative semidefinite ecl existing norm int norm uext existing control effort control effort uext uint fig comparison norm control effort uint input torque reference rigid body grouping together terms using fact ecl ecl hess tensor defined hess navigation function unique minimum exists compact neighborhood hess postivedefinite bounded implies therefore observed leading principal minors positive let shown ecl positive definite therefore choose ecl positive definite error dynamics dissipative sms control vector field proportional gradient navigation function result therefore achieves agas error dynamics agat control given uext acknowledgment part work carried second author visiting ravi banavar acknowledges pleasure support provided eferences vladimir arnol mathematical methods classical mechanics volume springer science business media ramaprakash bayadi ravi banavar almost global attitude stabilization rigid body internal external actuation schemes european journal control anthony bloch perinkulam krishnaprasad jerrold marsden alvarez stabilization rigid body dynamics internal external torques automatica francesco bullo andrew lewis geometric control ical systems modeling analysis design simple mechanical control systems volume springer science business media francesco bullo richard murray tracking fully actuated mechanical systems geometric framework automatica ecl yao cai qiang zhan path tracking control spherical mobile robot mechanism machine theory peter crouch spacecraft attitude control stabilization applications geometric control theory rigid body models ieee transactions automatic control sneha gajbhiye ravi banavar geometric approach tracking stabilization spherical robot actuated internal rotors corr christopher hall panagiotis tsiotras haijun shen tracking rigid body motion using thrusters momentum wheels astrodynamics specialist conference exhibit page shall express terms yury karavaev alexander kilin dynamics control set next choose spherical robot internal omniwheel platform regular chaotic dynamics makes let kkdi koditschek application total energy lyapunov function makes mechanical control systems contemporary mathematics taeyoung lee geometric tracking control attitude dynamics rigid body american control conference pages taeyoung lee melvin leoky harris mcclamroch geometric tracking control quadrotor uav decision control cdc ieee conference pages ieee twu madhushani dhs maithripala berg feedback regularization geometric pid control trajectory tracking coupled mechanical systems hoop robots inclined plane arxiv preprint sanjeeva maithripala jordan berg intrinsic robust pid controller lie groups ieee conference decision control pages ieee sanjeeva maithripala jordan berg wijesuriya dayawansa tracking simple mechanical systems general class lie groups ieee transactions automatic control jerrold marsden tudor ratiu introduction mechanics symmetry basic exposition classical mechanical systems volume springer science business media jerrold marsden scheurle reduced equations fields institute comm aradhana nayak ravi banavar tracking certain class simple mechanical systems arxiv preprint james ostrowski computing reduced equations robotic systems constraints symmetries ieee transactions robotics automation petersen graduate texts riemannian geometry springer october avishai weiss xuebo yang ilya kolmanovsky dennis bernstein spacecraft attitude control actuation aiaa guidance navigation control conference page yano integral formulas riemannian geometry volume dekker
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characterisation speech diversity using maps tom anderson david powers flinders university index terms kohonen speech typewriter australian speech som training boosting jan introduction report investigations speaker classification larger quantities unlabelled speech data using small sets manually phonemically annotated speech kohonen speech typewriter method comprised selforganising maps soms achieves low phoneme error rates som array cells learn vector representations data based neighbourhoods paper report method evaluate pronunciation using multilevel soms single syllable utterances study vowels following australian pronunciation vowel error rate shown table calculated comparing frequent vowel label outputted audio file vowel noted manual annotators method using initial vowel final submaps observed slightly better kohonen method using submaps disambiguate confused phonemes particularly chinese marginally significant table vowel error rate single classifier multisoms trained confused vowels vowels initial vowel final standard error parentheses based word tokens type single vowels hvd methodology used words austalk australian speech research corpus collected speakers australian english audio converted conventional cepstral coefficients mfccs log energy differential acceleration calculated windows offsets investigate sensitivity phonemic model dialect system trained using unlabelled speech one three different groups general australian speakers educated melbournian aged speakers chinese labelled using subset manually annotated data phonemes points three speakers follows base unit som trained using audio windows unlabelled data annotated data assigned units base map resulting labelled map used separating unlabelled input data submap boosting two approaches submapping three submap soms trained data segmented base map three maps subsets vowels shared confusion base map kohonen method three maps either vowels linguistic matlab scripts available authors request scripts mfccs modified melb chinese discussion goal experiments conducted work explore general capacity labelled data revealing characteristics dialect unlabelled speech chinesebackground australians educated melbournians australians generally results competitive original phonetic typewriter results demonstrating effectiveness mfcc vectors possibility improving using context future directions research interested characterising differences three groups acknowledgements austalk corpus collected part big asc project funded australian research council references results frequently confused phonemes first map identified using greedy method based high values confusion matrix base map vowel group hoyd horde vowel group head haired herd hid heared vowel group hud howd hade hard hide hod hode general kohonen neural phonetic typewriter computer vol mar cox acoustic characteristics vowels speech australian teenagers australian linguistics vol estival austalk corpus australian english proceedings international conference language resources evaluation reykjavik iceland see https details
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aug cluster algebras rings projections jiarui fei abstract let ring representations quiver suppose projects another quiver dimension vector exceptional representation show upper cluster algebra associated ice quiver upper cluster algebra associated obtained simple operations depending also study relation bases using quiver potential model introduction cluster algebra turned ubiquitous algebraic lie theory classical invariant theory realized motivation upper cluster algebra useful notion cluster algebra purpose series papers understand quiver representations upper cluster algebra use cluster structure solve problems invariant theory one family upper cluster algebras found ring complete triple flags introduced multigraded upper cluster algebras hive quivers grading also useful dimension graded piece counts certain coefficient hand many quivers projected triple quivers exceptional sequences fact author know quiver without oriented cycles projected triple quiver shall see paper cluster structure ring representations ultimately reduced case complete triple naturally generally one ask graded upper cluster algebra guca short might change another one degree weight restriction unable answer question general however grading restricted hyperplane cut weight provide complete solution additional conditions one special case particularly interesting projector graded upper cluster algebra projector simply means real case restriction leads deleting vertex general cases need restriction weights extended cluster general need delete mathematics subject classification primary secondary key words phrases cluster algebra ring quiver representation graded upper cluster algebra quiver potential weight restriction orthogonal projection vertex removal exceptional sequence cluster character polytope lattice point jiarui fei set frozen vertices freeze another set vertices let ice quiver associated upper cluster algebra let set vertices use notation quiver obtained deleting freezing respectively let quiver without oriented cycles dimension vector space representations acted product special linear groups natural base change rings invariant ring algebra graded sublattice equipped bilinear form denote component weight restriction applied leads change quiver dimension vector studied precise another quiver one vertex less dimension vector general weight restriction necessarily lead change however another special case called vertex removal additional condition denote new quiver dimension vector situation put together obtain two main results theorem theorem suppose naturally graded uca full rank real extremal grading cone theorem theorem suppose naturally graded uca full rank let vertex satisfying except set frozen vertices attached mutation quivers potentials invented model cluster algebras ice quiver potential associated jacobian algebra let set vectors proj introduced say iqp models algebra generic cluster character maps onto basis two main results say model sense hereditary two operations considered theorem theorem suppose projector models models theorem theorem assumption proposition suppose models models addition given lattice points polyhedron comprehensive application theory illustrated invite readers still provide many examples class examples explains get known cluster structure include total coordinate rings partial varieties unipotent subgroups gln second class examples mildly new rings complete whose cluster structure also obtained cluster algebras rings projections outline paper section recall graded upper cluster algebra following new emphasis grading given weight unimodular lattice section introduce crucial map projector prove precursors two main results proposition section recall theory ring quiver representations emphasis two operations orthogonal projection vertex removal give characterization projector semiinvariant ring proposition prove two main results theorem provide many examples section recall ice quivers potentials representations following recall generic cluster character current setting following section use restriction induction functor study relation cluster models two operations prove two main results theorem provide examples notations conventions vectors exclusively row vectors modules right modules quiver denote set vertices set arrows arrow denote tail head arrows composed left right path unadorned hom dim base write hom dim hom representation dimm dimension vector direct sum copies write instead traditional graded upper cluster algebras cluster algebras quiver directed graph possibly multiple arrows two vertices throughout assume quivers loops oriented ice quiver quiver vertices designated mutable rest designated frozen usually label quiver vertices mutable require arrows frozen vertices quiver uniquely determined matrix given combinatorial data cluster algebra completely encoded ice quiver definition let mutable vertex quiver mutation transforms new quiver via sequence three steps pair arrows introduce new arrow unless frozen case nothing reverse direction arrows incident remove oriented recipe reformulated terms follows let matrix obtained identity matrix replacing row vector jiarui fei write restriction upper left corner mutated related original one note quiver mutation involution preserves rank definition let containing seed pair consisting ice quiver together collection called extended cluster consisting algebraically independent elements one vertex elements associated mutable vertices called cluster variables form cluster elements associated frozen vertices called frozen variables coefficient variables seed mutation mutable vertex transforms seed follows new quiver new extended cluster new cluster variable replacing determined exchange relation note mutated seed contains variables original seed easy check one recover performing seed mutation two seeds obtained sequence mutations called denoted clear context may write definition cluster algebra cluster algebra associated seed subring generated elements extended clusters seeds note construction depends natural isomorphism mutation equivalence class initial quiver may drop simply write upper cluster algebras amazing property cluster algebras theorem laurent phenomenon element cluster algebra expressed terms extended cluster laurent polynomial polynomial coefficient variables since generated cluster variables seeds mutation equivalent theorem rephrased note slightly original one variables inverted cluster algebras rings projections definition upper cluster algebra upper cluster algebra uca short seed general may many seeds mutation equivalent useful test membership upper cluster algebra however following theorem allows many checking definition upper bounds theorem corollary suppose full rank particular remark theorem originally proved however carefully examine argument result also valid upper cluster algebra subring integral domain conventions however may fail noetherian since normality preserved localization intersection uca normal next lemma useful identify uca subalgebra given noetherian normal domain let extended cluster uca contained another commutative ring write union zero locus spec cluster variables definition say seed codimension short adjacent cluster codimension spec call two elements coprime codimension common zeros codimension spec expanding easily see codimension condition equivalent pair cluster variables pair coprime codimension lemma proposition let noetherian normal domain seed proof complement spec seed either original one adjacent one cluster variable vanish express laurent polynomial polynomial variables since conclude regular know corollary noetherian domain integral closure assumption conclude integral closure normality jiarui fei gradings let extended cluster vector write monomial set row matrix let suppose element written rational polynomial divisible assume matrix full rank elements algebraically independent vector uniquely determined call vector extended implies two elements set forms lemma lemma assume matrix full rank let subset distinct gvectors linearly independent let unimodular lattice equipped bilinear form bilinear form may neither symmetric must unimodular mapping homz isomorphism kind lattice one discussed throughout paper write left right orthogonal sublattices coxeter transformation unique linear transformation determined note via coxeter transformation allowing restrict discussion right orthogonal sublattices real vector two maps called right left projection definition weight configuration ice quiver assignment vertex weight vector mutable vertex mutation also transforms weight mutated quiver cluster algebras rings projections usually write weight pair mutation slight abuse notation view matrix whose row weight vector matrix notation condition equivalent zero matrix weight mutation iterated let weight unimodular lattice let vertex corresponds real vector left projection weight pair full subquiver obtained deleting vertex easy see projection commutes mutations away given weight assign multidegree weight upper cluster algebra setting deg mutation preserves multihomogeneousity say upper cluster algebra denoted refer graded seed note variables zero degrees homogeneous degree projections graded cluster algebras map let monoid simplicity assume submonoid inside unimdoular lattice let also integral domain assume noetherian graded pieces speak graded algebra graded subalgebra denote localization real weight algebra homomorphism following extension homogeneous components clearly surjective homomorphism uniquely extended quotient denote extension still let embedding definition element called projector real also called projector projector maps onto call projection lemma projector factors proof clearly proposition suppose projector following dim unit irreducible jiarui fei ufd contains units extremal cone proof projector dim let projector suppose one say must hand domain assumption unit suppose extremal irreducible contradicts ufd property cases throughout paper use vertex cluster variable potential confusion abuse notation map always quotient extended cluster say laurent polynomial always implies polynomial variables omit part important context goal subsection relate another upper cluster algebra real lemma reduces general case case frozen freezing set vertex mean make frozen let quiver obtained freezing lemma suppose full rank let proof show subalgebra theorem show written laurent polynomial note written laurent polynomial polynomial exchange polynomial substituting get required laurent polynomial since cluster also cluster element written laurent polynomial necessary polynomial inverting obtain conclude remark lemma still holds full rank condition replaced condition noetherian seed indeed seed also seed lemma still remark applies several results depending lemma proposition corollary theorem corollary graded seed since algebraically independent seen weight graded seed lemma sequence mutations proof show rather clear commutes deletion map commutes mutations homomorphism cluster algebras rings projections corollary extended cluster extended cluster particular proof suppose obtained sequence mutations claim since mutation away lemma proposition suppose frozen full rank let proof lemma need consider case frozen show let extended cluster since frozen extended cluster lemma homogeneous write laurent polynomial equal let another extended cluster write laurent polynomial see factor power lies image let homogeneous let extended cluster choose extended cluster corollary write laurent polynomial apply map laurent polynomial thus obtain containment remark saw corollary however examples show assumption proposition inclusion strict linear function linear isomorphism ker iee construction map xge let matrix linear transform lemma frozen associated proof show columns equal equivalently since required identity obtained taking sides row applying equation get jiarui fei let linear map forgetting coordinate left inverse xee say upper cluster algebra basis whose elements distinct see section corollary suppose frozen full rank basis moreover projector well proof suppose basis element written lemma xee proposition basis suppose projector remain show chosen distinct basis since elements homogeneous basis ker show injective restricted ker ker ker last statement follows cases necessary general cases necessary much complicated obtain similar result imposing restriction however follows need assume real use instead definition set vertices called attached another set vertices resp whenever resp pair form quiver deleting freezing let set vertices write set cluster variables cluster also write restriction proposition let full rank suppose weight except set frozen vertices graded subalgebra equal attached proof argument lemma shows contained thus indeed theorem show written laurent polynomial note written laurentqpolynomial inqx polynomial let exchange polynomial thus get required laurent polynomial conversely homogeneous write laurent polynomial given extended cluster polynomial variables assume obtained mutations away contained clear cluster mutated seed since attached still see variables appear otherwise remain show polynomial write laurent polynomial mutated cluster since attached exchange polynomial contains variables particular sign decompose sum homogeneous cluster algebras rings projections laurent monomials cancelations see must degree monomial otherwise must polynomial remark similar remark replace full rank condition condition noetherian seed also seed lemma still remark applies several results depending proposition corollary theorem theorem corollary remark analogous remark assumption proposition general situation proposition evaluation left inverse embedding similar lemma let linear maps proof following corollary similar easier corollary corollary suppose situation proposition well moreover projections rings rings quiver representations let recall rings quiver representations let quiver without oriented cycles algebraically closed characteristic zero dimension vector let vector space write component space representations hom product general linear group acts natural base change interested rings ring weight space decomposition jiarui fei runs multiplicative characters refer decomposition recall character weight vector det let understand multihomogeneous components since oriented cycles degree zero component projective presentation view element homotopy category proj bounded complexes projective representations view weight element grothendieck group proj concretely suppose use notation indecomposable projective representation corresponding vertex let weight projective presentation weight constructed necessary element construction refer readers original text section fact theorem span base field let set weights known weight corresponds dimension vector euler form euler form induces form space weights satisfying unimodular lattice form used weight later next theorem easy consequence king stability criterion proposition theorem general representation mean small zariski open subset depends context following use notation mean general representation subrepresentation theorem theorem particular pointed saturated monoid recall dimension vector called real root called schur homq general real schur root dense orbit lemma lemma real schur root lemma lemma suppose exact sequence representations hdiml hdimn function scalar equal minimal presentation hdiml hdimn cluster algebras rings projections orthogonal projections two dimension vector write homq extq general recall exceptional sequence dimension vector sequence real schur roots called quiver called complete quiver quiver exceptional sequence quiver vertices labeled extq arrows according theorem extq given exceptional sequence dimension vector general consider right orthogonal subcategory rep according abelian subcategory equivalent module category another quiver denoted say projects case unique quiver exceptional sequence concatenation two sequences complete exceptional sequence refer construction linear isometry respect euler forms conversely linear map left inverse composition map given recursion maps induce linear map weights still denoted write compute later use formulae theorem theorem let real schur root proposition let real root extremal projector proof theorem ufd since pointed direction follows proposition conversely let show fact schur general representation decomposes lemma must since extremal weights must lie real clearly impossible dimension vector must homq indeed let rank general morphism general representation general representation note claim observe must multiple say otherwise lemma weights ker lie contradicting extremal exact sequence exceptional representation dimension applying homq jiarui fei homq since real schur general representation dimension isomorphic nontrivial homomorphism thus construct homomorphism image strictly containing must theorem function weight linear combination minimal presentation general representation lemma factor conclude projector lemma say ring naturally graded upper cluster algebra isomorphism theorem suppose naturally graded uca full rank real extremal schur naturally graded uca proof proof proposition seen must schur rest follows proposition theorem remark follows always identify linear isometry available may keep instead writing remark note exceptional sequence extremal extremal theorem inductively applied exceptional sequence complete flag varieties unipotent subgroups sextuple flags recall main result consider triple quiver standard dimension vector indicated let eai unit vector supported vertex arm let weight given use convention zero vector ean clear extremal proved ring guca ice hive quiver ice hive quiver size vertices arranged triangular grid ice quiver plane label vertices shown figure note three vertices missing boundary vertices frozen three types arrows easy see full rank example complete varieties consider exceptional sequence note order cluster algebras rings projections figure ice hive quiver matter unique quiver exceptional sequence completing given quiver displayed corresponds vertices respectively arrows vertices denote quiver sfn let dimension vector given verify embedding dimension vector goes hard see sfn total coordinate ring fln complete variety fln vector space let divisor class group variety multiplication given multiplying homogeneous sections rational functions however rigorously prove fact need variation git write lecture notes observe exceptional sequence corresponds last row frozen vertices let ice quiver obtained deleting last row frozen vertices see figure follows theorem proposition total coordinate ring fln guca new weight computed example clearly extremal sfn example unipotent subgroups sketch get cluster structure maximal unipotent subgroup gln previous example jiarui fei figure ice quiver left right leave details readers although ungraded algebra polynomial ring certainly multigrading taken account verify corresponds real schur root denoted fact exceptional sequence sfn unique quiver exceptional sequence completing given verify arrow matrix strict matrix entries equal one let dimension vector verify embedding dimension vector goes clearly coordinate ring let ice quiver obtained deleting right left row frozen vertices follows proposition coordinate ring guca want tell curious readers project vertex except none weights corresponds real schur root example complete sextuple consider exceptional sequence unique quiver exceptional sequence completing given order matter quiver sextuple quiver eai correspond vertices left right three respectively corresponds central vertex call quiver sextuple quiver denoted let dimension vector indicated verify embedding dimension vector goes let ice quiver obtained deleting vertices corresponding follows theorem cluster algebras rings projections figure ice quiver proposition ring guca also easily obtained since appear proposition list readers convenience let similar eir ejr ekr vertex removal let quiver dimension vector vertex let quiver vertex removal algebraic morphism sending lemma assume following condition min vertex removal induces embedding mapping onto runs weights supported proof show let min mout second fundamental theorem invariant theory identify hom min hom mout rank maps hom min mout due assumption maps hom min mout rank equal jiarui fei words lemma weight corresponding unit vector follows proposition lemma theorem suppose naturally graded uca full rank let vertex satisfying except set frozen vertices naturally graded uca attached finally remark proposition also used remove arrows outer automorphism group quiver linear space spanned arrows particular arrow associated torus action commutes action torus induces another grading ring cluster seed homogeneous respect grading induced use proposition remove arrow partial flag varieties quintuple quadruple flags triple flag exercise partial varieties able cluster structure total coordinate ring partial varieties type since divisor class group grassmannian total coordinate ring usual coordinate ring hence able recover result well start result example remove vertices long arm sfn depending particular variety following hint may useful let sequence vertices contains mutable vertices row want remove vertices apply sequence mutation following order able apply theorem exercise complete quintuple quadruple get cluster structure complete quintuple start example remove vertices order remove vertices another get complete quadruple shall proposition ring guca ring guca going give precise construction quivers rather clear quiver displayed conjecture quiver snm projected triple quiver ring snm complete upper cluster algebra rest subsection illustrate single example inductively apply vertex removal technique theorem cluster algebras rings projections figure quiver left right possible cluster structure method described fails believe cluster structure may exist however method deterministic method suppose guca want remove vertex satisfying let assume schur root stabilizer general position action krull dimension given dim dim compute vertex removal theorem works vanish exactly frozen vertices vertices choice otherwise method fails cluster structure may exist next need seek suitable sequence mutations application mutable sequence exists method fails cluster structure may exist succeeds take attached example example explain cluster algebra structure ring obtained removing three arms two arms also explain cluster algebra structure may exist forget either third arm suggest readers play example keller applet directly apply theorem remove one say arm results correspond figure apply another mutation make quiver easier draw step optional next remove arm mutate required delete freeze perform optional mutation similarly remove second arm mutate delete freeze perform optional mutation finally remove arm mutate delete freeze perform optional mutation results correspond figure interested readers verify using types cluster algebras jiarui fei last one cluster variable associated vertex weight fij one verify correspond real schur roots ring graded upper cluster algebra however want remove third arm unable choose set expected cardinality even worse matter mutate least vertices suspect may upper cluster algebra cluster characters quivers potentials quivers potentials mutation quivers potentials invented model cluster algebras emphasis ice quivers considered arrows frozen vertices following potential ice quiver possibly linear combination oriented cycles precisely potential element trace space completion path algebra closure commutator subspace pair ice quiver potential iqp short arrow cyclic linear extension derivative potential jacobian ideal closed ideal generated jacobian algebra quotient algebra polynomial completion unnecessary case throughout paper key notion introduced mutation quivers potentials decorated representations ice quiver nondegenerate potential see mutation certain sense lifts mutation since need mutation explicit way refer readers original text definition decorated representation jacobian algebra pair rep let rep set decorated representations isomorphism bijection two additive categories rep proj mapping representation minimal presentation rep simple representation suppose corresponds projective presentation cluster algebras rings projections definition decorated representation weight vector image proj representation called min vertices definition iqp freezing set vertices operation set vertices frozen delete arrows frozen vertices also keep potential remark slightly delete arrows frozen vertices thus change potential well ignored look category representations see reason two two categories equivalent objects take value cluster character however insist deleting arrows frozen vertices cause additional trouble constructing two functors section definition decorated representation called supporting vertices mutable denote repg set decorated representations cluster character definition cluster character repg gre variety parameterizing quotient representations denotes topological turns lemma repg element note theorem theorem suppose iqp full rank let set decorated representations distinct maps bijectively set linearly independent elements upper cluster algebra however clear whether uca basis distinct definition associate reduced presentation space phomj homj vector satisfying max denote coker cokernel general presentation phomj reader aware coker notation rather representation write coker simply means take general presentation phomj let cokernel definition weight vector proj called coker let set vectors proj definition generic character gre coker jiarui fei known lemma general presentation phomj corresponds decorated representation image also contained particular remark assume situation theorem moreover assume graded components let set forms basis lemma definition say iqp models algebra generic cluster character maps onto basis upper cluster algebra simply say cluster model addition given lattice points polyhedron say model polyhedral require iqp graded components remark cluster model equivalent iqp models restriction induction definition iqp deleting set vertices operation delete vertices set incoming outgoing arrows zero denote new iqp operation deleting restricting vertex set follows shall mainly consider case single vertex general case easily treated induction according proposition restriction induces algebra epimorphism denote let rese rep rep restriction scalar functor induced concretely rep extension zeros quiver clearly representation representation rese conversely denote functor rep rep inde functor inde left adjoint exact functor rese extend functor rese decorated representations copying decorated spaces denote map still rese rep rep functor inde proj inde proj naturally extended note indecomposable projective rep inde linear map forgetting coordinate lemma rese differ coordinate moreover rese always proof recall dim coker dim dim refer readers section map comparing rese changes three terms right last statement also clear equation cluster algebras rings projections lemma composition inde rese identity functor particular frozen coker coker inde proof since composition right exact show identity indecomposable projective representations lemma rep following presentation mpe rese apply right exact functor inde exact sequence get inde inde inde rese frozen rese last statement follows suppose pvj view pui row vectors entries pui pvj represented matrix aji aji linear combination paths description see functor induces surjective algebraic map homj homje particular general enough homj inde take value generic character decorated representation corresponding orthogonal projection focus case frozen keep assumption real lemma following commutative diagram frozen repg inde rep proof surjectivity follows proposition repg rese repg frozen recall lemma inde rese rese isomorphic rese restricting subquiver gre lemma gre hand inde since xee gre conclude inde remark projector embedding however want warn readers another functor rese embedding commutative diagram general rese theorem suppose projector models models jiarui fei proof assume frozen show cwe forms basis assumption subset repg distinct ker remark forms basis moreover remark end section element chosen take value generic character since projector proposition inde forms basis since inde similar argument corollary see elements inde distinct come back general situation necessary frozen claim spans spans always large coker mee supported mee equal note also mee since spans lemma finally apply previous conclusion remark model polyhedral model totally unimodular similar remark also applies theorem corollary suppose models full rank real weight extremal models proof follows theorem proposition theorem example continued quiver equipped rigid potential see detail proved models ring moreover given lattice points hive cone follows maximal straight path must start frozen vertex end another frozen vertex strict subpaths maximal straight path subpaths include trivial paths frozen notation stand path definition given weight quiver convex polyhedral cone necessarily bounded convex polytope cut hyperplane sections follows see lattice points counts refer readers partitions example example continued corollary given lattice points polyhedral cone denoted similar calculation section cone also described calculation similar proposition dimension graded piece sfn counts kostka number multiplicity irreducible complex representation gln tensor cluster algebras rings projections product partitions weight determined follows alternatively one see theorem sfn kostka interpreted certain interpretation proposition kostka number counted lattice points example example continued corollary given lattice points polyhedral cone denoted since ring polynomial ring variables see cone generated extremal rays moreover cone also described denote hyperplane representation ghn cone related kostant vector partition function type follows recall counts lattice points polytope matrix form rows positive roots type easy verify linear algebra ghn since totally unimodular two polytope number lattice points question recall kostant formulae relates kostka number alternating sum values partition function steinberg formulae relates alternating sum kostka numbers dot action half sum positive roots type see two formula relations among polytopes example example continued corollary given lattice points polyhedral cone denoted similar calculation section cone also described similar proposition dimension graded piece counts certain multiple section determined weight similar proposition multiple coefficients dinary coefficient counted lattice points jiarui fei vertex removal suppose attached graded seed restrict seed get new graded seed section denote recall situation proposition epimorphism proof following lemma theorem corollary completely analogous easier lemma theorem corollary leave details readers lemma following commutative diagram assumption proposition repg resv rep theorem assumption proposition suppose models models corollary assumption theorem suppose ring modeled modeled going provide examples part good examples illustration contained section references atiyah macdonald introduction commutative algebra publishing reading mills ont berenstein fomin zelevinsky cluster algebras iii upper bounds double bruhat cells duke math derksen fei general presentations algebras adv math derksen weyman quivers saturation coefficients amer math soc derksen weyman combinatorics quiver representation ann inst fourier grenoble derksen weyman zelevinsky quivers potentials representations selecta math derksen weyman zelevinsky quivers potentials representations amer math soc domokos zubkov quivers determinants transform groups fei cluster algebras rings triple flags fei cluster algebras invarant theory kronecker coefficients fock goncharov moduli spaces local systems higher theory publ math inst hautes fomin pylyavskyy tensor diagrams cluster algebras fomin zelevinsky cluster algebras foundations amer math soc fomin zelevinsky cluster algebras coefficients compos math geiss leclerc auslander algebras initial seeds cluster journal london mathematical society geiss leclerc partial flag varieties preprojective algebras ann inst fourier grenoble cluster algebras rings projections humphreys introduction lie algebras representation theory graduate texts mathematics vol new keller quiver mutation java http king moduli representations algebras quart math oxford ser knutson tao honeycomb model gln tensor products proof saturation conjecture math soc plamondon generic bases cluster algebras cluster category int math res notices popov vinberg invariant theory algebraic geometry encyclopaedia mathematical sciences vol berlin schofield quivers london math soc schofield general representations quivers proc london math soc speyer infinitely generated upper cluster algebra schofield van den bergh quivers arbitrary dimension vectors indag math scott grassmannians cluster algebras proc london math soc noncommutative localization noncommutative geometry localization algebra topology london math soc lecture note cambridge univ press cambridge national center theoretical sciences taipei taiwan address jrfei jiarui fei remove remove remove remove remove remove remove figure operations towards steps
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partial reinitialisation optimisers dec ilia zintchenko theoretical physics station zurich eth zurich zurich switzerland zintchenko itp phys ethz matthew hastings station microsoft research santa barbara usa mahastin microsoft com quantum architectures computation group microsoft research redmond usa nathan wiebe quantum architectures computation group microsoft research redmond usa nawiebe microsoft com ethan brown theoretical physics station zurich eth zurich zurich switzerland ebrown itp phys ethz matthias troyer theoretical physics station zurich eth zurich zurich switzerland troyer phys ethz abstract heuristic optimisers search optimal configuration variables relative objective function often get stuck local optima algorithm unable find improvement standard approach circumvent problem involves periodically restarting algorithm random initial configurations improvement found propose method partial reinitialization whereby attempt find better solution variables rather whole configuration much information gained previous runs hence retained leads significant improvements quality solution found given time variety optimisation problems machine learning introduction multivariate optimisation central importance industry used range problems finding solutions satisfy set constraints training classifiers machine learning despite wide range plications nearly involve finding optimal assignment set variables respect cost function exact global optimum however rarely required general significantly costly find good local optimum heuristic optimisers therefore ubiquitous many industrial application problems zhang likas burges heuristic optimiser generally starts random configuration variables optimises configuration local optimum found convex optimisation problem optimum also global optimum effort required however problems considered general exponentially many local optima hence probability particular local optimum also global one exponentially suppressed nevertheless problems often contain amount structure makes solution useful structure used obtain configurations given cost much faster random guessing subsequent query cost function makes use preceding queries make informed guess optimal configuration partial reinitialisation optimisers figure illustration partial reinitialisation algorithm level variables reinitialised black circles bottom level optimiser called yellow circles level starts optimal configuration parent reinitialises variables calling optimiser next level green arrows checkpoints optimal configuration found kept level flow left right blue arrows strategy frequently used escape local optimum restart optimisation process new random initial configuration repeating process multiple times local optimum lowest cost returned high probability better certainty worse initial configuration although restarts allow optimiser get local optima different restarts also completely decoupled information learned structure problem one restart passed next relearned scratch hence way running optimiser effectively coarse grained random guessing guess improved towards local optimum introduce general approach address problem allows optimiser find global optimum high probability single run algorithm lets assume heuristic picks variables also let define optimiser configuration returns reinitialising variables typical smaller found heuristic get configuration local optimum optimiser would return configuration however reinitialising variables may allow optimiser find set new local optima may worse better current one starting lets increase better local optimum reachable typical picked heuristic current local optimum good number better local optima negligible increasing would reduce chance finding better local optimum hence except beginning optimisation process optimiser higher chance finding better local optimum rather variables variables reinitialised optimiser called configuration becomes high probability chance finding better local optimum decreases prevent optimiser getting stuck variables turn get optima variables repeating process iteratively time increasing size configuration becomes global optimum high probability process hence refine local optimizer global optimizer call approach partial following algorithm see also fig algorithm partial reinitialisation input current level number reinitialisations number variables reinitialisation call heuristic optimiser else reinitialise variables call partial reinitialisation level end cost cost end end levels hierarchy algorithm started mth level configuration denoted checkpoints checkpoints optimal configurations found thus far algorithm reverts improved configuration found partial reinitialization level reinitialisations variables performed complexity single run algorithm hence number variables also heuristic picks variables important simplest approach pick variables random however partial reinitialisation optimisers variables chosen according heuristic probability given size leads optimal configuration maximised one approach pick optimality variables within set depends values variables set much possible could increase chance getting local optimum reducing number constraints rest system outcome heuristic optimiser directly depend initial configuration also random seed could used optimise variables within variables problem kept fixed approach employed zintchenko finding ground states ising spin glasses simulated annealing showed significant speedup relative conventional global restarts samples needed conclude high probability value small hand unstructured search number requisite reinitialisations exponential number variables thus value taken makes implicit assumption structure complexity problem benchmarks set machine learning problems study advantage partial reinitialisation compared standard full reinitialisation finding optimum model parameters problems picked ones experience size problem chosen large enough finding optimal configuration using respective standard algorithms worth noting machine learning local optimum sometimes sufficient even desired global optimum due overtraining lecun however show partial reinitialisation improves quality final solution also speed local optium given quality obtained also although general expensive overtraining reduced using classicifation accuracy random training data cost function mean standard deviation distribition hence either fully reinitialise subsets variables add small perturbations variables combine two partially perturb variables could improve performance simplicity use benchmarks simplicity use one level hierarchy full reinitialisation calling heuristic optimiser full reinitialisation multiple reinitialisations variables perfomed maintain generality choose random benchmark problems optimisation problem space continuous variables concept partial reinitialisation extended partially reinitialising variable addition variables rather setting variable random value within domain perturbed example adding noise standard deviation choosing although chosen empirically optimise performance optimiser set randomly chosen optimization problems also theoretical basis behind choice chosen ensure probability local optima respect reinitiasation variables input maximised call configuration probabilistically require probability reinitialization variables improve optimum less assert probability given reinitialisation improves objective function suffices take donmez dln thus values specify value needed conclude local optimum probabilistically optimal within margin error furthermore consider constant logarithmic number parameters benchmarks size subset denoted number partial reinitialisations denoted done within full reinitialisation manually optimised true optima respective performance metrics performance measure benchmark use optimal cost obtained given elapsed time number iterations averaged multiple runs different random initial states elapsed time measured intel xeon processors training hidden markov models learning temporal patterns signal central importance wide range fields including speech recognition finance bioinformatics classic method model systems hidden markov models hmm based assumption signal follows markov process future state system depends solely present state without memory past partial reinitialisation optimisers wcss log time time figure learning random sequences using baumwelch algorithm median observe sequence within model elapsed training time runs different random seeds shown full reinitialisations blue partial reinitialisations red figure benchmark set clu median sum squares wcss elapsed time runs algorithm shown blue curve full reinitialisations red curve partial reinitialisations assumption turns surprisingly accurate many applications starting random transition emission matrices number hidden states least large length model trained observed certainty within model use two visible states number hidden states bits input generality impose restrictions transition matrix discrete hmms consider system one possible states hidden observer starting discrete probability distribution states time evolves system transition states according probability matrix hidden state emmit one possible visible states model hence composed three parts initial probability distribution length hidden states transition matrix hidden states emission matrix hidden state possible visible states training given input sequence matrices optimised maximise likelihood sequence observed standard algorithm training hmms baumwelch algorithm rabiner based procedure computes posterior marginal distributions using dynamic programming approach model commonly initialised random values optimised maximise expectation input sequence convergece local optimum improve accuracy multiple restarts usually performed sequence restarts investigate partial reinitialisation improve convergence rate towards global optimum benchmark choose learning random string although artificial problem simple free parameters time finding model accurately represents sequence full reinitialisation starts random emission transition matrices partial reinitialisation elements matrices corresponding hidden states reinitialised found reinitialisations variables within global restart optimal leads much rapid increase accuracy training time full reinitialisations see fig clustering dividing objects clusters according similarity metric central importance data analysis employed ubiquitously machine learning given set points space idea assign points clusters way maximise similarities within cluster minimise similarities clusters similarity defined many ways depending particular needs application use euclidean distance one widely used algorithm finding clusters algorithm macqueen kmeans searches assignment points clusters minimise sum square distances center seeks minimise following partial reinitialisation optimisers cost function kxj affinity propagation benchmark problem set standard clustering data set clu clusters points sampled gaussian distributions points around uniformly placed centres full reinitialisation starts intial assignment centers random points using forgy method forgy partial reinitialisation single cluster center reinitialised found partial reinitialisations global restart optimal except beginning given quality clusters obtained significantly quicker partial rather full reinitialisation see fig reason full reinitialisation efficient finding low quality clusters regime cost high larger strides reducing cost made random guessing optimising bad global restart partial reinitialsiations however cost threshold becomes harder reach partial reinitialisation quickly becomes efficient clustering often robust approach called clustering data pick best cluster center one points cluster rather euclidean center standard algorithm finding clusters partitioning around medoids pam theodoridis koutroumbas similar algorithm pam iterates assigning points closest center finding best center cluster convergence centers constrainted positions points cluster assignment occur twice run trivial upper bound number iterations convergence points clusters similar however typically converges quickly wcss starting random initialisation centers iteration proceeds two stages first points assigned nearest cluster center second part center picked euclidean center cluster repeated convergence local optimum cost never increased convergence guaranteed shown exists problems converges exponentially many iterations number points vattani smoothed running time however polynomial arthur typically converges quickly similar algorithm multiple global reinitialisations centers often performed improve quality clusters time figure benchmark problem clustering images faces olivetti face database fac clusters affinity propagation green curve compared kmedoids full blue curve partial red curve reinitialisation affinity propagation benchmark coresponding timings done using web interface aff benchmark problem use clustering images faces olivetti database fac dataset used previously benchmark novel method called affinity propagation frey dueck finding clusters pam repeat benchmark also compare pam partial reinitialisations chose find clusters roughly regime affinity propagation largest advantage frey dueck within gloal restart found reinitialisations single randomly chosen cluster center optimal use similarity matrix available fac entries squared distances images affinity propagation indeed quickly finds much better clusters pam full reinitialisations however time pam partial reinitialisations finds even better clusters keep improving even time see fig full reinitialisation finds better cluters beginning similar reasons training bolzmann machines boltzmann machines class highly generalizable models related neural networks proven useful modeling data sets many areas including speech vision bengio hinton salakhutdinov hinton goal boltzmann machine training replicate relative error mean training objective function partial reinitialisation optimisers trial number trial number figure training restricted bolzmann machine rbm left absolute values objective functions right relative difference mean oml approximate global optima found maximizing oml experimental runs ability distribution set training data rather identify patterns data set generalize cases yet observed boltzmann machine takes following form let define two layers units call visible layer hidden layer visible units comprise input output boltzmann machine hidden units latent variables marginalized order generate correlations present data call vector visible units vector hidden units units typically taken binary joint probability configuration visible hidden units exp normalization factor known partition function energy matrix weights models interaction pairs hidden visible units vectors biases units model viewed ising model complete bipartite graph thermal equilibrium model commonly known restricted boltzmann machine rbm rbms stacked form layered boltzmann machines sometimes called deep boltzmann machines simplicity focus training rbms since training deep boltzmann machines using popular methods contrastive divergence training involves optimising weights biases layered rbm independently training process involves optimising maximum likelihood training objective oml oml regularization term introduced prevent overfitting exact computation training objective function hard means computation expected intractable large rbms reasonable complexity theoretic assumptions although oml efficiently computed derivatives efficiently estimated using method known contrastive divergence algorithm described detail hinton uses markov chain algorithm estimates expectation values hidden visible units needed compute derivatives oml specifically hvi idata hvi imodel denotes expectation value gibbs distribution visible units clamped training data denotes unconstrained expectation value derivative respect biases similar locally optimal configurations weights biases calculated stochastic gradient ascent using approximate gradients benchmark examine small synthetic examples boltzmann machines training objective function calculated exactly although differs task based estimates performance bolzmann machine focus contrastive divergence partial reinitialisation optimisers training algorithm formally seeks approximate gradients objective function value training objective function natural metric comparison purposes classification accuracy appropriate data set training set consists four distinct functions bnv otherwise mod bitwise negations order make data set challenging learn add bernoulli noise training examples one hundred training examples used instance learning rate epochs per reinitialization take model rbm consisting visible units hidden units finally rather reinitialising fixed number weights iteration update weight fixed probability formally differs previous approaches performs similarly reinitialising constant fraction upon initialisation variable drawn gaussian distribution see figure partial reinitialization accelerates process estimating global optima contrastive divergence training find optimal probability reinitializing weight bias roughly tends lead substantial reductions number reinitializations needed attain particular objective function particular variables reset stage takes resets average obtain training objective function strategy resets every variable attains resets furthermore see evidence suggests possible polynomial advantage strategy traditional random resetting strategy strong regularization expected lead much simpler landscape optimisation since optimisation problem becomes convex find difference partial total reinitisation strategies becomes order although partial reinitialising variables continues superior method small differences could also arise solely stochastic nature contrastive divergence training fact takes fewer epoches partial reinitialization approach local optima evident benchmarks partial reinitialization substantially reduce complexity finding approximate global optimum boltzmann machines consequently expect approach also work optimising boltzmann training large based problems mnist digit classification ilarly improvements able leveraged forms neural net training feedforward convolutional neural nets conclusion introduced general purpose approach termed partial significantly improve performance optimiser numerically explored comparisons state art algorithms range optimisation problems machine learning although used basic version algorithm single additional level hierarchy picking variables random advantage standard full reinitialisation substantial expect hierarchy multiple levels picked according advanced heuristics lead even improvements acknowledgments would like thank supernova inspiration imriska katzgraber kosenkov soluyanov svore wecker fruitful discussions paper based upon work supported part odni iarpa via mit lincoln laboratory air force contract views conclusions contained herein authors interpreted necessarily representing official policies endorsements either expressed implied odni iarpa government government authorized reproduce distribute reprints governmental purpose notwithstanding copyright annotation thereon references affinity propagation web app http accessed clustering datasets http accessed olivetti face database http accessed olivetti face database http accessed arthur david manthey bodo heiko kmeans polynomial smoothed complexity proceedings annual ieee symposium foundations computer science focs washington usa ieee computer partial reinitialisation optimisers ciety isbn doi url http bengio yoshua learning deep architectures foundations machine learning burges christopher svore krysta marie bennett paul pastusiak andrzej qiang learning rank using ensemble models yahoo learning rank challenge donmez pinar svore krysta burges christopher local optimality lambdarank proceedings international acm sigir conference research development information retrieval acm forgy cluster analysis multivariate data efficiency versus interpretability classification biometrics frey brendan dueck delbert clustering passing messages data points science url affinitypropagation hinton geoffrey training products experts minimizing contrastive divergence neural computation lecun yann bengio yoshua hinton geoffrey deep learning nature may issn url http insight likas aristidis vlassis nikos verbeek jakob global clustering algorithm pattern recognition macqueen methods classification analysis multivariate observations url http rabiner lawrence tutorial hidden markov models selected applications speech recognition proceedings ieee salakhutdinov ruslan hinton geoffrey deep boltzmann machines international conference artificial intelligence statistics theodoridis sergios koutroumbas konstantinos chapter clustering algorithms iii schemes based function optimization koutroumbas sergios theodoridiskonstantinos pattern recognition third edition academic press san diego third edition edition isbn doi http url http vattani andrea requires exponentially many iterations even plane discrete computational geometry issn doi url http zhang lintao madigan conor moskewicz matthew malik sharad efficient conflict driven learning boolean satisfiability solver proceedings international conference design ieee press zintchenko ilia hastings matthew troyer matthias local global ground states ising spin glasses phys rev jan doi url http
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sep modules irreducible justin chen youngsu kim abstract show graded submodule noetherian module written proper intersection graded submodules written proper intersection submodules generally show natural extension index reducibility graded setting coincides ordinary index reducibility also investigate question uniqueness components decomposition well relation index reducibility ideal largest graded subideal let ring commutative submodule recall said irreducible whenever definition let ring graded graded submodule said whenever graded equivalently iff whenever finite collection graded submodules follows directly definitions graded submodule irreducible natural question ask whether converse holds answered first result theorem let ring noetherian graded graded submodule irreducible iff analogy recall case monomial ideals polynomial ring textbook exercise monomial ideals irreducible iff written proper intersection monomial ideals iff generated pure powers variables section moreover decompositions ideals unique note possible even monomial ideal intersection ideals field char however monomial case quickly resolved following lemma lemma monomial ideal minimal generator relatively prime monomials contrast formula case mention general graded ring far worse behaved polynomial ring mathematics subject classification justin chen youngsu kim giving proof theorem introduce numerical invariant graded submodule noetherian every graded submodule finite intersection submodules maximal counterexample would definition would intersection two strictly larger graded submodules maximality finite intersections thus well contradiction motivates following definition definition let noetherian submodule index reducibility min irreducible graded graded index reducibility min module understood may simply write noetherian hypothesis guarantees always finite moreover iff irreducible graded iff local primary maximal ideal definition index reducibility appeared literature although best knowledge graded index reducibility appeared vector space dimension socle residue field see addition graded local ring hard show indices coincide lemma however case polynomial ring graded index reducibility general vector space dimension base field one must compute ranks laurent polynomial ring instead definitions alone clear relation holds general somewhat surprisingly always equal fact follows theorem indeed following three statements equivalent noetherian graded module theorem submodule irreducible graded submodule graded submodule finite intersection irreducible graded submodules key feature statement simultaneous requirements finiteness irreducibility gradedness notice finiteness along either irreducibility gradedness easy satisfy although independent proof statement first sight may seem follow directly noetherian hypothesis would give another proof theorem far unable find one proof theorem begin proof theorem reductions use hereafter without mention definitions modules irreducible graded also submodules graded annr also graded henceforth pass factor ring assume ring noetherian graded prime ideal graded ring graded rmodule set set homogeneous elements graded finally refer sections notation basic results graded case lemma let noetherian ring prime ideal finitely generated submodule irreducible iff irreducible addition graded iff proof may assume ass consists nonzerodivisors thus localization map injective conversely submodules extended submodules graded case proof applies homogeneous localization remark despite elementary nature proof lemma conditions quite delicate general irreducibility preserved faithfully flat ring extensions localizations example irreducibility source extend target take field completion hand domain spec flat injective rrp rrq artinian case terms next give formula resp socle rank resp graded socle rank provide proof graded case proof holds verbatim local case removing appearances word graded lemma let noetherian ring finitely generated submodule local artinian dimk rankk proof notice graded field graded free direct sums twists replacing may assume set decomposition irredundant graded module iff indecomposable iff every graded submodule thus decomposition implies also homr homr graded module case parameter ideal local ring see satz justin chen youngsu kim since assr structure theorem injectives implies indecomposable one rankk rankk homr rankk homr rankk homr rankk homr rankk homr homr remark let noetherian local ring type finitely generated defined type dimk extdepth since homr lemma implies artinian type recall local ring gorenstein iff type remark together lemmas yield following irreducibility criterion ideal noetherian ring irreducible iff primary generically gorenstein recall ideal generically gorenstein gorenstein ass lemmas along one last trick yield theorems mentioned theorem let ring noetherian graded graded submodule irreducible iff proof direction clear suppose replacing may assume take graded primary decomposition graded primary prop hypothesis primary assr graded prime ideal lemma artinian local ring lemma implies twist localizing yields prp lemma rmp lemma irreducible next show equivalence mentioned introduction theorem let ring noetherian graded following statements equivalent every submodule irreducible every graded submodule every graded submodule finite intersection irreducible graded submodules proof take decomposition modules length irredundant irreducible decomposition exercise every decomposition length modules irreducible graded irreducible iff iff iff let graded submodule write assumption irreducible let take decomposition irreducible graded since irreducible examples example let noetherian ring radical ideal min ass number minimal equivalently associated primes fail satisfies serre condition following example shows example let field char set ideals generated regular sequences hence irreducible remark since gorenstein thus irredundant irreducible decomposition note graded graded theorem well indeed irredundant decomposition analogy monomial case interesting ask extent ideals unique addressed remark see observe artinian local ring socle soc finally local ring implies particular iff ideal example interesting properties summarize following remark remark minimal following ways minimal length among graded finite length monomial dim minimal among polynomial rings graded primary reducible contained principal ideal component irredundant decomposition gradedirreducibles unique furthermore component must form one along forms irredundant decomposition justin chen youngsu kim proof let graded finite length length isomorphic respectively length isomorphic either monomial dim every graded irreducible hand either complete intersection hence irreducible remark contained principal ideal let ideals recall first show contains form degree suppose since least degrees among minimal generating set must contain observe complete intersection ideal therefore contradiction next show without loss generality generated forms degree write certainly generated forms degree since graded would equal suppose generated forms degree dimk denotes graded piece would imply contradiction thus without loss generality generated degree say linear form form degree since exist forms fact else contradiction hence also generate order contain must divide hence greatest common divisor therefore one show generated forms degree since suppose implies conclude let linear form equivalent since conclude proves first part second statement remains show either reasoning shows suffices show hilbert functions agree follows since maximal proper subspace contains forms degree relationship thus far started graded object seen graded ungraded notions irreducibility agree graded objects end briefly discussing different setting namely starting object passing closest graded approximation submodule graded module let denote submodule generated graded elements let prime ideal noetherian graded ring ideals although distinct often differ slightly various properties invariants example modules irreducible height height resp gorenstein iff finitely generated graded type type vein natural ask compares ideal answer following special case proposition let noetherian graded ring prime ideal principal particular irreducible iff irreducible lemma let ring maximal ideal type proof let unique homogeneous prime ideal let maximal ideal artinian local result follows remark otherwise largest graded subideal type type rrq equalities follow theorem remark proof lemma respectively notice artinian local ring proof first notice since hypothesis numbers change upon going modulo homogeneously localizing hence may assume local unique maximal homogeneous ideal maximal ideal since ass principal ideal generated nonzerodivisor say type type artinian local ring lemma type isomorphism lemma homr conclude type dimk dimk homrp type hypothesis necessary notice graded implies graded conversely following examples computed help example let field deg deg deg ideals ideals next example demonstrates principal stronger condition generated elements example let field deg deg deg let however ideal requires least two generators instance observe homogeneous minimal generating set lift part minimal generating set every minimal generating set justin chen youngsu kim however even general condition generated elements necessary conclusion hold even simplest case prime example moh primes let field char fix odd consider ring map tnm ker height prime ideal height graded prime ideal ufd hence principal however moh shown requires least generators thus conditions numbers generators unlikely necessary view examples pose following question question let noetherian graded ring prime ideal necessary conditions acknowledgements project started authors met ams sectional meeting san francisco state university october independently reading question answered positively theorem thank professor david eisenbud professor bernd ulrich advice encouragement professor william heinzer reading early draft also thank referee helpful comments references bourbaki chapitres hermann paris winfried bruns herzog rings volume cambridge studies advanced mathematics cambridge university press cambridge david eisenbud commutative algebra volume graduate texts mathematics new york view toward algebraic geometry shiro goto naoyoshi suzuki index reducibility parameter ideals local ring algebra daniel grayson michael stillman software system research algebraic geometry available http wolfgang irreduzible ideale kommutativen ringen math herzog takayuki hibi monomial ideals volume graduate texts mathematics london london ezra miller bernd sturmfels combinatorial commutative algebra volume graduate texts mathematics new york moh unboundedness generators prime ideals power series rings three variables math soc japan northcott irreducible ideals local rings london math department mathematics university california berkeley california address jchen department mathematics university california riverside california address
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commensurability artin groups raags defined trees diameter mar montserrat ilya kazachkov alexander zakharov abstract paper study classification artin groups commensurability characterise commensurability classes raags defined trees diameter particular prove conjecture behrstock neumann infinitely many commensurability classes hence give first examples raags commensurable introduction context one basic problems locally compact topological groups classify lattices commensurability recall two lattices commensurable exists finite index particular commensurable lattices covolumes commensurable real numbers rational ratio notion commensurability generalized better suit topological geometric properties compare groups without requiring subgroups common group precisely say two groups abstractly commensurable isomorphic finite index subgroups article concerned notion abstract commensurability simply refer commensurability mentioned commensurability closely related geometry group indeed finitely generated group endowed natural welldefined since finitely generated group finite index subgroups follows commensurable groups gromov suggested study groups geometric point view understand relation two concepts precisely basic problem geometric group theory classify commensurability classes perhaps within certain class finitely generated groups understand whether classes coincide key words phrases artin groups commensurability work supported erc grant basque government grant grant ministerio economia competitividad spain first author supported juan cierva programme spanish government third author supported russian foundation basic research project cmup funded fct portugal national mec european structural funds feder partnership agreement kazachkov zakharov classification groups commensurability abstract classical case long history number famous solutions diverse classes groups lie groups hyperbolic groups groups groups etc see instance paper focus question classification artin groups raags short commensurability recall raag finitely presented group described finite simplicial graph commutation graph following way vertices bijective correspondence generators set defining relations consists commutation relations one pair generators connected edge raags become central group theory study interweaves geometric group theory areas mathematics class interpolates two classical families groups free free abelian groups study provides uniform approaches proofs well rich generalisations results free free abelian groups study class different perspectives contributed development new rich theories theory cat cube complexes essential ingredient agol solution virtually fibered conjecture commensurability classification raags previously solved following classes raags free groups free abelian groups free products free groups free abelian groups tree diameter connected graph without degree one vertices outer automorphism finite induced turns inside class raags classification commensurability coincides classification known cases rigidity results mainly consequence rigid structure intersection pattern flats universal cover salvetti complex paper describe commensurability classes raags defined trees diameter describe minimal group commensurability class minimal terms number generators rank abelianization particular show exist infinitely many different commensurability classes confirming conjecture behrstock neumann paper authors show raags defined trees diameter least provide first examples raags commensurable classical case lattices locally compact topological groups define ordered set plays role covolume prove groups commensurable commensurability raags sets commensurable cardinality constant rational ordered ratios main results let simplicial graph denote raag defined commutation graph call vertices graph canonical generators purposes convenient encode finite trees diameter four follows let finite tree diameter four let path without backtracking length four one leaf another definition contains vertices let middle vertex immediate see choice vertex depend choice path length four call center leaf connected edge called hair vertex vertices connected edge hair called pivots finite tree diameter uniquely defined number hair vertices number pivots given degree hence encode finite tree diameter positive integers integer moreover either indeed diameter see figure figure tree diameter given tree diameter four denote set numbers original usage two real numbers commensurable ratio rational fashion say two ordered sets commensurable cardinality exists quotient kazachkov zakharov case write say set minimal greatest common divisor gcd clear commensurability class set exists minimal set belongs class namely mdk show commensurability class set determines commensurability class raag defined tree diameter theorem characterisation commensurability classes let two finite trees diameter let consider sets commensurable commensurable denote path vertices edges minimal raag class mean raag minimal number generators defined graph minimal number vertices among commutation graphs raags theorem let finite tree diameter let commensurability class let minimal raag belongs either raag defined tree minimal commensurability class raag defined path diameter results extend naturally commensurability classification coxeter groups recall every raag embeds naturally finite index subgroup coxeter group say see hence following result corollary infinitely many commensurable coxeter groups defined graphs diameter cliques dimension strategy proof discussed previous results commensurability raags structure intersection pattern flats inside universal cover salvetti complex rigid geometry determines forces commensurability see case trees diameter geometry sufficient determine commensurability classes however use structure intersection pattern flats together algebra derive result broad strokes strategy follows two given raags defined trees diameter associate linear system equations show commensurable system positive integer solutions see section study system determine conditions trees system positive integer solutions allows conclude corresponding raags commensurable see sections order obtain characterisation prove conditions satisfied system positive integer solutions exhibit isomorphic finite index subgroups conclude commensurable see section believe general strategy proof reduce existence subgroups linear system equations positive integer solutions used study commensurability commensurability raags classes raags defined trees general raags real obstacle elucidate necessary conditions system equations positive integer solutions linear algebra problem basics raags section recall preliminary results raags introduce notation use throughout text let finite simplicial graph graph without loops multiple edges vertex set edge set artin group raag short defined commutation graph group given following presentation whenever elements called canonical generators let full subgraph hard show see instance raag subgroup generated let word alphabet denote element corresponding also say word represents denote length word word called geodesic minimal length among words representing element raags word transformed geodesic representing element applying free cancellations permutations letters allowed commutativity relations moreover two geodesic words representing element transformed applying commutativity relations see details let geodesic word representing length defined length element called cyclically reduced equivalently length minimal conjugacy class every element conjugate cyclically reduced one say letter occurs word least one letters either given element denote alph set letters occurring geodesic word representing depend choice due remarks also define subgroup generated letters occur geodesic word represents commute subgroup independent choice due remarks note also letter commutes commutes every letter alph see paper always conjugate follows raag define graph follows complement graph graph union connected components induce decomposition direct product kazachkov zakharov given cyclically reduced set alph consider graph alph full subgraph vertex set consisting letters alph graph connected call block alph connected decompose product wjt number connected components alph word wji word letters connected component clearly words wjt pairwise commute word wji block refer expression block decomposition element called least root simply root exists positive integer exist case write result raags least roots root element defined uniquely every next result describes centralisers elements raags since centralizers conjugate elements conjugate subgroups suffices describe centralizers cyclically reduced elements theorem centraliser theorem theorem let cyclically reduced block decomposition centraliser following subgroup following two corollaries follow immediately theorem definitions corollary corollary vertex centralizer generated vertices star vertices adjacent particular tree centralizer vertex isomorphic degree isomorphic leaf contains free group otherwise following corollary play key role paper corollary let tree let conjugate power vertex generator case free group proof one implication follows immediately corollaries implication suppose centralizer let cyclically reduced since follows theorem cyclic since tree implies power generator definition corollary centralizers elements raags raags proof follows immediately theorem fact raag definition note general raags contain lots subgroups raags even finite index commensurability raags necessary conditions commensurability reduced extension graph centraliser splitting section recall notions reduced extension graph reduced centraliser splitting particular case underlying graph tree reduced extension graph see goal section show exists equivariant isomorphism reduced extension graph tree reduced centraliser splitting definition extension graph see let raag underlying commutation graph extension graph defined follows vertex set set elements conjugate canonical generators vertices two vertices joined edge corresponding group elements commute group acts conjugation sense extension graph encodes structure intersection pattern flats inside universal cover salvetti complex plays essential role establishing rigidity class raags finite outer automorphism groups see observe raags split fundamental groups graph groups whose vertex groups centralisers vertex generators paper work centraliser splitting defined follows definition reduced centraliser splitting let tree let raag underlying graph centraliser splitting graph groups defined follows graph splitting isomorphic vertex group every vertex defined centralizer corresponding vertex generator note vertex vertices adjacent free group rank see corollary particular abelian degree case contained centralizer vertex adjacent edge connecting vertices edge group note centralizer splitting reduced since every vertex degree vertex group equal incident edge group thus makes sense consider reduced centraliser splitting tree obtained centralizer splitting removing vertices degree splitting vertex groups edge groups isomorphic particular splitting already reduced show lemma centraliser splitting corresponds extension graph reduced centraliser splitting corresponds reduced extension graph define definition reduced extension graph tree define reduced extension graph full subgraph extension graph whose vertex set denote set elements conjugate canonical generators corresponding vertices degree exactly centralizers reduced extension graph play essential role classification raags commensurability subgroup finite index intersects cyclic subgroup associated vertex groups reduced extension graph cyclic subgroup kazachkov zakharov hagi aki hand case trees description centralisers raags every element whose centraliser belongs cyclic subgroup aki hagi see corollary since set elements centralisers invariant set isomorphism follows reduced extension graph algebraic invariant class finite index subgroups graph whose vertex set correspondence maximal cyclic subgroups generated elements centraliser edge whenever corresponding elements commute isomorphic reduced extension graph see lemma observation one deduce many classes raags commensurable without using stronger fact instance raags whose defining graphs trees commensurable raags whose defining graph cycles raags defined cycles different lengths commensurable etc point couple remarks order firstly observation extend extension graph commensurable raags may isomorphic extension graphs instance show raags defined paths length commensurable extension graphs isomorphic leaves tree also leaves extension graphs minimal distances extension graphs leaves respectively reason elements abelian centralisers powers conjugates canonical generators accounted extension graph secondly isomorphism reduced extension graphs necessary condition far sufficient raags defined trees diameter isomorphic reduced extension graphs infinitely many different commensurability classes among next lemmas notice two trees reduced extension graph tree associated reduced centraliser splitting equivalent reduced extension graph quotient action invariant isomorphism lemma tree extension graph also tree isomorphic tree corresponding centralizer splitting moreover reduced subtree isomorphic tree corresponding extension graph reduced centralizer splitting graph isomorphisms equivariant corresponds natural action sense action conjugation tree centralizer splitting reduced centralizer splitting respectively denote proof every vertex form canonical generator theory vertices correspond left cosets centralizers canonical generators define morphism sending vertex form gives bijection vertex sets definition two vertices connected edge iff iff iff commensurability raags iff theory since tree therefore appearing two vertices connected edge iff iff shows indeed graph isomorphism equivariant gives equivariant isomorphism restriction commensurability invariants reduced extension graph quotient graph goal section show reduced extension graph quotient graph defined commensurability invariants suppose finite tree finite index subgroup acts restriction let acts reduced extension graph quotient graph goal section show commensurable reduced extension graphs isomorphic quotient graphs note natural projection graph morphisms note also graph finite since finite index defined even fact since finite index exists natural lemma let vertex vertex bijection vertices adjacent equivalence classes adjacent modulo conjugation vertices bijection edges incident equivalence classes adjacent modulo conjugation vertices proof first claim follows immediately definition quotient graph incident let ends let two different edges respectively project edge exists takes since sure belong different orbits action even happens takes leaves fixed means thus second claim also holds particular graph multiple edges cycles loops suppose finite trees denote suppose commensurable thus exist finite index subgroups isomorphism defined actions conjugation consider reduced extension graphs respectively finite graphs defined kazachkov zakharov lemma group isomorphism induces graph isomorphisms proof let vertices degree vertices degree canonical generators centralizers canonical generators centralizers notice elements centralizers conjugates powers element since finite index finite index elements centralizers conjugates powers element belong elements set uki analogously elements centralizers conjugates powers element belong elements set thus isomorphism take set set follows first define vertex set let define morphism trees vertex one since finite index exists minimal positive integer centralizer also centralizer thus positive integer equal one denote let note also equal minimal positive integer since also proper power impossible since proper power extend edges natural way indeed description centralisers raags adjacent iff see theorem two vertices take positive integers iff iff adjacent shows morphism trees moreover fact isomorphism trees since inverse morphism constructed way follows vertex restrict graph isomorphism orbit projections proves vertex edge lemma commensurability invariant linear relations minimal exponents far observed isomorphisms leave set powers conjugates canonical generators centralisers invariant however isomorphism sends priori relation integer numbers corresponding powers commensurability raags example let hci hck hck clearly maps isomorphism taken arbitrarily hand consider hck hck hci assume isomorphism sends power conjugate either power conjugate get constraint possible powers image indeed isomorphism induces isomorphism subgroup hence index since among conjugates powers generators minimal exponent assumption also furthermore since sum minimal exponents conjugates fixed generator equal index case since upto relabeling two covers degree one conjugates powers generators minimal exponent follows either sent conjugate generator exactly two conjugates generator sent square generator conjugate minimal exponent generator subgroup next goal formalise generalise ideas order find linear relations minimal exponents different conjugacy classes powers generators belong subgroups correspondingly strategy follows already established isomorphism finite index subgroups induces isomorphism centralisers conjugates generators correspondingly since finite index centralisers respectively note form group hence isomorphism induces isomorphism images free groups obtained killing centres see lemma general conjugate canonical generator minimal exponent belongs coincide minimal exponent belongs quotient center hwi homomorphism see example refer minimal quotient exponent equal minimal positive number see definition since ranks isomorphic subgroups free groups schreier formula relates corresponding indexes show lemma index coincides sum minimal quotient exponents conjugates given generator belong goal section describe linear relation minimal quotient exponents conjugates generators belong correspondingly see corollary section establish relation minimal exponents minimal quotient exponents see lemma deduce linear relation minimal exponents conjugates generators belong correspondingly see lemma kazachkov zakharov linear relations minimal quotient exponents fix finite index subgroup encode minimal exponent label vertex reduced extension graph definition label minimal exponent let vertex thus also element define label vertex denoted minimal positive integer number exists since finite index similarly encode minimal quotient exponent label edge reduced extension graph follows edge connecting vertices define label edge vertex denoted minimal positive integer exists integer note always suppose analogously label defined note definition edges conjugation note labels vertices edges invariant action indeed iff iff means define labels quotient graph well vertex define label vertex denoted label vertex analogously edge connecting vertices define label edge vertex denoted label end edge labels note labels vertices edges positive integers example minimal exponent minimal quotient exponent consider free group rank consider index subgroup let hci let finite index subgroup easy computation shows minimal exponent equals min let natural projection hence minimal quotient exponent equals min let thus canonical generator let vertex centralizer suppose vertices adjacent degree degree centralizer note ugk hwi cyclic free group rank basis ugk let free group rank basis denote factorization center homomorphism induced map ugi induces isomorphism ugk speak cycles schreier graph mean simple cycles edges labelled generator mean schreier graph respect generators commensurability raags lemma notation epimorphism induces group embedding finite index subgroup proof let minimal positive power belongs note since positive integer every element center also commute also ker hwi ker hwj means induces embedding since finite index follows finite index lemma notation following statements hold induces bijection edges incident cycles schreier graph correspond generators every edge connecting vertex label respect equal length cycle schreier graph proof consider cayley graph respect basis line cayley graph path every edge labelled generator incident lines cayley graph define bijection edges labelled one generators centralizers let one generators edge connecting proof lemma means note defined multiplication right element centraliser suppose since commutes particular well yxg word obtained replacing ugi thought element ugk let line cayley graph passing vertex corresponding edges labeled element equal one elements note indeed suppose write two ways one thus considered equality free group means means line cayley graph defined correctly follows definition surjective show injective suppose connects anf connects thus shows bijective thus also note acts set edges incident vertex acts set action trivial induces action set edges incident hand acts kazachkov zakharov cayley graph natural way left multiplication also acts set lines cayley graph defined recall induces isomorphism follows definition every incident every edge denote actions defined dots means bijection induces bijection orbits edges incident action one side orbits lines cayley graph action side lemma orbits edges incident action correspondence edges incident orbits lines cayley graph action natural correspondence cycles schreier graph labelled generators defines desired bijection proves first claim lemma incident turn second claim let edge let connect recall label equal label minimal positive integer exists integer note means since ker hwi follows equal minimal positive integer let denote line passing edges labelled thus equal minimal positive integer equivalent thus equal minimal positive exponent points line cayley graph equivalent action exactly length corresponding cycle schreier graph proves second claim orbit lemma notation let two vertices action natural isomorphism restricts isomorphism respects bijections lemma proof let let hug uhg huhg also ugi uhg let isomorphism define isomorphism definitions commensurability raags note induces isomorphism thus also induces isomorphism schreier graphs natural way recall bijection edges incident cycles schreier graph correspond generators analogously bijection edges incident cycles schreier graph correspond generators follows definition every edge incident cycle goes cycle isomorphism induced respects bijections vertex lemma different choices give groups subgroups well bijections lemma isomorphism means exact choice unimportant abusing notation denote vertex vertex lemma notation vertex let vertex degree connected edge edges incident vertex connects vertex lvi edges incident connects vertex lwi number vertices degree greater adjacent proof according lemma bijection set edges set cycles schreier graph correspond conjugates length cycle equal corresponding edge label sum lengths cycles equal number vertices schreier graph equal thus follows summing equalities form vertices degree greater adjacent gives suppose finite trees commensurable exist finite index subgroups isomorphism defined recall vertex vertex finite index subgroups defined lemma kazachkov zakharov lemma let vertex let corresponding vertex graph isomorphism lemma let suppose degree degree think element let proof let positive integer note thus particular schreier index formula implies combining two previous lemmas establish linear relation minimal quotient exponents record following corollary corollary let vertex let let suppose degree degree let edges incident connects vertex let let also number vertices adjacent number vertices adjacent lwi proof since graph isomorphism see lemma edges incident lemma lwi statement follows lemma linear relations minimal exponents recall definitions labels vertices edges see definition incident vertex lemma let edge respectively denote labels vertices denote label edge respect denote label edge respect analogously let edge incident vertex denote labels vertices respectively denote label edge respect denote label edge respect commensurability raags lemma let edge connecting vertices positive integer ends proof let edge rewrite definition minimal positive integer exists integer note positive integer exists integer indeed denote let greatest common divisor exist integers since commute since gcd means thus recall minimal positive integer thus positive integer positive integer analogously remains show indeed suppose means let follows shown thus means analogous argument shows thus lemma let edge connecting vertices positive integer proof already know lemma positive integers thus suffices show kazachkov zakharov ends let edge let also rewrite recall minimal positive integer exists integer fix number denote element power note note also definition labels shown proof lemma thus power roots raags unique see section definition means argument roles interchanged shows thus proves lemma lemmas imply labels edges satisfy following equations lemma suppose vertex graph isomorphism lemma suppose degree adjacent vertices degree degree adjacent vertices degree let edges adjacent suppose labels commensurability raags proof let edge connect corollary lvi multiplying sides obtain lvi multiplying sides obtain lvi according lemma lvi lvi lvi lvi thus imply definition type vertex vertex say vertex type vertex note vertex type adjacent vertex type adjacent adjacent infinitely many commensurability classes raags defined trees diameter turn attention proofs main results first show infinitely many different commensurability classes inside class raags defined trees diameter particular case theorem give separate proof two reasons first one technically easier helps warm general case secondly reader interested qualitative aspect result suffices read theorem denote tree diameter pivots one degree degree hair vertices see figure thus theorem infinitely many commensurability classes let suppose groups commensurable particular infinitely many commensurability classes among raags defined trees diameter fact see next section commensurable kazachkov zakharov figure tree diameter pivots proof theorem let center vertex center vertex let pivots degrees correspondingly pivots degrees correspondingly let finite index subgroups lemma lemma applied system equations labels vertices edges show system equations positive integer solutions unless note could exist vertices following nine types see definition also follows definition following statements hold every vertex type connected vertices every vertex connected vertices type every vertex connected vertices every vertex connected vertices thus two cases either vertices type vertices type case suppose first vertices type let vertex vertices adjacent type let edges incident let connect possibly coincide lemma notations let vertices every vertex consider edges incident finish vertices fixed commensurability raags let xij sum edges ends let xij edges follows lemma sum equalities form vertices fixed group summands vertex label corresponding vertex together obtain summing using get analogously summing using get multiplying using obtain since definition every follows note apply argument roles interchanged thus get means desired kazachkov zakharov case suppose vertices type let vertex vertices adjacent type let edges incident let connect possibly vertices coincide lemma notations obtain let vertices every vertex consider edges incident finish vertices note edges always exist let sum edges ends analogously consider edges incident finish vertices note edges always exist let sum edges ends let follows lemma every sum equalities form vertices group summands vertex label corresponding vertex together obtain analogously summing equalities form vertices get together equalities imply note apply argument roles interchanged thus get follows particular proves theorem criterion raags defined trees diameter prove main result paper theorem let suppose commensurable commensurable moreover commensurable commensurability raags proof theorem first prove first claim let central vertex central vertex let pivots pivots let finite index subgroups lemma lemma applied system equations labels vertices edges show system equations solutions positive integers unless commensurable always mean ranges ranges unless stated otherwise note could exist vertices following types see definition also follows definition following statements hold every vertex connected vertices every vertex connected vertices every vertex connected vertices every vertex connected vertices thus two cases either vertices vertices case suppose first vertices denote vertices vertices short let vertex vertices adjacent type let edges incident let connect possibly vertices coincide lemma notations obtain suppose vertices every vertex consider edges incident finish vertices fixed let xij sum edges ends let yij sum edges ends let xij yijq edges follows lemma xilq every xkl kazachkov zakharov analogously denote ykj every yklq sum equalities form vertices fixed group summands vertex label corresponding vertex together obtain xij xij possible denote xij xij xij get xij possible analogously sum equalities form vertices fixed group summands vertex label corresponding vertex together obtain yijq yijq possible denote yij yijq yij yijq get yij yij possible get xkl commensurability raags possible summing equalities get following system equations analogously get ykl ykl possible summing equalities get following system equations let aij matrix rows columns aij vertices aij otherwise note every xij yijq vertices type adjacent obviously every vertex type adjacent least one vertex least one means xij xij xij vertices type aij analogously yij yij yij yij yij vertices type aij note also every exist vertices analogously every exist vertices exist means matrix zero rows columns combining following equations conditions xij yij yij aij xkl xkl kazachkov zakharov ykl ykl ykl xij yij yij aij xij yij yij xij xij without loss generality assume denote lower integer part even odd claim notation aij provided proof prove claim induction dij dij min thus interested case dij base induction case dij either case included inductive step suppose claim proved want prove aij provided aij provided case conditions means equation following form xrr equation following form also last two lines equations obtain xrr since implies xrr equality obtained also implies commensurability raags equality obtained multiplying using obtain xrr note since otherwise row zero impossible implies apply analogous arguments induction hypothesis equation following form yrr yrr equation following form also last two lines equations obtain yrr yrr since implies yrr yrr equality obtained also implies equality attained multiplying using obtain yrr yrr yrr note yrr since otherwise column zero impossible implies kazachkov zakharov combined gives inequalities also turn equalities mentioned means combined induction hypothesis proves aij provided proves claim thus proved aij provided claim notation one proof suppose contrary suppose first even aij aij particular taking get since pairs since pairs zero row contradiction odd follows aij aij particular denote due means among equations following xbb xbb xbb immediately implies contradiction shows follows aij identity matrix since zero rows columns according xij xii commensurability raags therefore turns xkk xkk immediately implies finishes case case suppose vertices proof case almost proof case theorem let vertex vertices adjacent types let edges incident let connect possibly vertices coincide lemma notations obtain suppose vertices every vertex consider edges incident finish vertices note edges always exist let zjq sum edges ends zjq every let follows lemma every sum equalities form vertices group summands vertex label corresponding vertex together obtain analogously summing equalities form vertices get together equalities imply contradiction finishes proof first claim theorem proof second claim similar easier indeed let apply arguments proof first claim case argument shows still holds contradiction proof case kazachkov zakharov proves theorem characterisation commensurability classes raags defined trees diameter turn prove commensurability raags defined trees diameter proposition let exist integer finite index subgroup tree diameter form particular commensurable exists finite index subgroup proof first consider case let pivots let epimorphism induced map canonical generator different hard see ker isomorphic see suppose denote pivot valency central vertex let let free group basis natural epimorphism fix without loss generality assume always suppose since otherwise claim obvious let leaves adjacent pivot consider finite index subgroup defined subgroup corresponding finite cover bouquet circles defined follows akl mean simple cycle length vertices appearing order akl analogously cycles take two cycles first one form akl length edges labelled second one form length edges labelled identify cycles vertex vertex basepoint based cover construct degree finite cover bouquet circles construct new vertices added edges first complete constructed graph cover free group adding loops labelled vertices loops added adding loops labelled vertices akl done construction cover consider two cases first case suppose second case assume first case every add cycle akl labelled every add loops labelled vertices aki add cycle aki aki akl length edges labelled see figure adding loops labelled generators vertices graph obtain finite cover bouquet circles hence defines finite index subgroup commensurability raags figure constructing cover case figure constructing cover case second case every following first add cycle edges labelled add loops labelled vertices aki finally add cycle aki aki akl length labelled see figure adding loops labelled generators vertices graph obtain finite cover bouquet circles hence defines finite index subgroup continue assumptions second case proof first case analogous note index construction epimorphisms free group basis construction image index hence follows index choosing maximal subtree spanned edges labelled labelled using nielsen transformations difficult see free basis consists following generators elements form free basis kazachkov zakharov pki pki pkl generators words containing least two cpl indeed seen direct calculations applying schreier formula subgroup index free group rank schreier formula generators first three types formula follows thus finite index subgroup generated set note elements become equal consists one generator show note group splits fundamental group star groups leaves vertex groups leaves equal vertex group center vertex equal edge groups reduced centralizer splitting let tree corresponding splitting recall theory vertices correspond left cosets vertex groups edges correspond left cosets edge groups consider finite subgraph tree consists vertex incident edges going vertices edges pjl going vertices pjl together vertices note vertices indeed different thus graph star leaves claim fundamental domain action contains exactly one representative vertex edge orbit action first show contain two vertices edges orbit action suffices show vertices note vertex since even suppose vertex form exists definition see figure follows contradiction definition commensurability raags get contradiction way way prove vertex pjl vertex thus indeed contain two vertices edges orbit action show contains least one representative orbit fundamental domain note suffices show every edge incident edge taken edge element let edge incident vertex theory gei suppose contain suppose case analogous definition see figure set coset representatives gei qei since definition edge qei done edge incident vertex theory gei bei suppose contain definition bei definition case edge incident vertex pjl analogous proves fundamental domain action let induced splitting fundamental group graph groups corresponding induced action let underlying graph let natural projection morphism follows morphism restricted induces isomorphism graphs vertex group equal stabilizer vertex every vertex means star leaves center vertex vertex group hgenerators type vertex groups gui leaves following ipl edge groups corresponding intersections vertex groups since follows vertex groups direct products free group infinite cyclic group moreover since construction conjugates belong follows every edge group direct product cyclic group generated subgroup let let pivots valency let leaves adjacent pivot denote center hairs kazachkov zakharov follows following map induces isomorphism psl words means reduced centralizer splitting conclude proposition let two trees diameter exists finite index subgroup particular commensurable proof let since group group correspondingly retracts onto free subgroup correspondingly generated pivots hair vertices full preimage subgroup correspondingly finite index index subgroup correspondingly define subgroups via covers bouquet circles correspondingly subgroup defined follows take points every pivot generator add cycle length labelled generator complete obtained graph cover adding loops labelled hair generators every vertex subgroup defined similar fashion note free subgroups index correspondingly rank define subgroups full preimages subgroups correspondingly index index claim groups isomorphic indeed since rank one see free basis includes powers pivot generators well elements similar similar proof proposition follows split fundamental groups star groups leaves induced splittings respect reduced centralizer splittings respectively splittings isomorphic reduced centralizer splitting claim lemma holds commensurability raags theorem let two trees diameter suppose group commensurable proof propositions suffices prove statement case case consider first case proof second case analogous let consider homomorphism induced map center tree canonical generator let kernel similarly let homomorphism induced map center tree canonical generator let kernel theory difficult see nml mnl since nmi mni follows hence commensurable turn attention description minimal elements commensurability classes first record raags defined paths length commensurable fact new mentioned koberda proposition commensurable proof let let homomorphism defined map set ker clear index subgroup let homomorphism defined map let ker leaves pivots correspondingly clear index subgroup straightforward application technique shows isomorphic deduce following results theorem characterisation commensurability classes let two finite trees diameter let consider sets commensurable commensurable proof consequence theorem theorem corollary groups commensurable kazachkov zakharov theorem minimal raag commensurability class let finite tree diameter let commensurability class let minimal raag belongs either raag defined tree minimal commensurability class raag defined path diameter proof consequence proposition theorem references bass degree polynomial growth finitely generated nilpotent groups proc london math soc behrstock januszkiewicz neumann commensurability classification free products finitely generated abelian groups proc math soc behrstock januszkiewicz neumann classification high dimensional artin groups groups geom dynamics behrstock neumann classification graph manifold groups duke math burger mozes lattices product trees inst hautes tudes sci publ davis januszkiewicz artin groups commensurable coxeter groups pure appl algebra deligne mostow commensurabilities among lattices annals mathematics studies princeton university press duchamp krob partially commutative magnus transformations internat algebra comput esyp kazachkov remeslennikov divisibility theory complexity algorithms free partially commutative groups contemp amer math providence garrido abstract commensurability group groups geom dyn grigorchuk wilson structural property concerning abstract commensurability subgroups london math soc gromov groups polynomial growth expanding maps inst hautes tudes sci publ gromov geometric group theory vol asymptotic invariants infinite groups lond math soc lecture notes huang commensurability groups raags karrass pietrowski solitar finite infinite cyclic extensions free groups australian math soc kim koberda embedability artin groups geom topol kim koberda geometry curve graph artin group int algebra comput margulis arithmeticity nonuniform lattices funkcional anal schwartz classification rank one lattices publ math inst hautes etudes servatius automorphisms graph groups algebra siegel symplectic geometry amer stallings groups infinitely many ends ann math whyte coarse bundles wise curved squared complexes aperiodic tilings finite groups princeton university commensurability raags ikerbasque basque foundation science matematika saila sarriena leioa bizkaia spain address montsecasals ikerbasque basque foundation science matematika saila sarriena leioa bizkaia spain address matematika saila sarriena leioa bizkaia spain russian foreign trade academy pudovkin street moscow russia address
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jun dynamic stripes exploiting dynamic precision requirements activation values neural networks alberto delmas patrick judd sayeh sharify andreas moshovos department electrical computer engineering university toronto juddpatr sayeh moshovos deep neural network dnn accelerator uses computation offer performance proportional precision activation values precisions determined priori using profiling selected per layer granularity paper presents dynamic stripes extension stripes detects precision variance runtime finer granularity extra level precision reduction increases performance stripes ntroduction previously described stripes accelerator exploited variable precision requirements deep learning neural networks improve performance energy efficiency previously disclosed design hardware expected prior processing layer precision required layer would communicated software per layer precisions thus adjusted runtime reflect additional reduction precision may possible layer even smaller granularity however underlying compute units capable exploiting precisions much finer granularity layer described implementation chip comprised tiles processing filters weights synapses per filter set activations broadcast tiles one bit per cycle layer precision activations positions significant least significant bits msb lsb respectively adjusted per layer however precision could easily adapted smaller granularity example precision could adjusted per group activations processed concurrently per group activations broadcast column sips described implementation similar techniques dynamic precision reduction computation used signal processing best knowledge first application technique neural networks implified xample figure shows detection done example group four activations originally expressed bits precision without mechanism detect precision runtime could precision used process group activations would bits since precisions determined prior profiling run whole layer would case example activation possibly position profiling runs determined needed able take value another example would case two activations profiling run determined needed able take values respectively accuracy maintained desired however run activations take values depend given input network prior inputs case example recurrent neural networks figure shows specific values group four activations represented using bits among four values highest bit position appears position counting whereas lowest bit position appears position figure shows network calculating set signals orj one per input bit position orj signals calculated using set cascaded gates figure shown diamonds detection block accepts orj signals input performs leading bit simply leading detection identify position result block encoding since information communicated processing tiles encoder block used shown convert encoding number binary function offset encoder detection uses identical block block priority orj inputs reversed effect calculates trailing bit detector figure shows example detecting set neuron values since input neurons use bits two offsets encoded using bits process group neurons dispatcher send starting offset units decrement offset every subsequent cycle dispatcher signal last cycle processing group current offset becomes equal done extra wire signals end processing neuron group assuming processing starts bit position counter keeps track current bit position broadcast comparator sets end group signal arrive example four neurons processed concurrently single group practice may neurons one group case group neurons processed corresponding neuron lanes made wait neuron lanes finish advancing next group neuron values alternatively dispatcher synapse buffer modified support per neuron group accesses expense additional area memory bandwidth group assuming processing starts bit position counter keeps track current bit position broadcast comparator sets end group signal units arrive figure shows example dynamic stripes organization specifically dispatcher communicates activation bits precision info processing tiles configuration activations grouped subgroups activations subgroup dispatcher sends current offset along end group eog signal assuming baseline precision bits offset require wires eog one total extra wires per subgroup activations groupings possible fig example illustrating runtime detection precision necessary process group activations fig dynamic stripes overview dispatcher processing titles iii esign previous section considered case four activations forming single group processed concurrently implementations may choose process activations multiple groups example implementation described stripes activations processed concurrently broadcast tiles tile processes filters weights per filter total weights per tile tile comprises grid serial units sips sips along row process set weights sips along column process set activations accordingly implementation precision specified dynamically adjusted whole group activations processed concurrently determined separately group activations processed concurrently column sips determining precision necessary group activations performed dispatcher prior communicating activations units processing neurons sent units bit assuming neurons positive transposer first calculates logical bits position nib applies leading bit detector determine highest bit position bit appears similarly transposer uses trailing bit detector determine lowest bit position bit appears process neurons transposer sends along bits also offset via set extra wires additional wire indicates end processing neuron modified serial unit figure shows implementation modified serialinner product unit additional shifter appears output adder tree shifter control signal allows output adder tree adjusted right bit position accumulated running sum kept accumulator msb neuron shift neg synapse synapse neg max msb prec fig modified sip valuation erformance evaluated performance improvement possible dynamic stripes set modern image classification convolutional neural networks table shows resulting performance improvements equivalently configured dadiannao accelerator performance measurements obtained via simulator extended support dynamic detection activation precisions set results first set precisions per layer network alexnet nin googlenet vgg vgg vgg geomean str dadn example activation value activation converted whereas would converted either threshold activation would represented exactly representation different floating point representation experimented optimization found greatly improves performance alexnet specifically method followed get precision profiles via gradient descent determine number required accuracy alexnet gives opposed using fixed number bits take significant every neuron precision profile obtained step run pragmatic performance dadiannao becomes pragmatic configuration tiles remain dispatcher needs modified stop transmitting oneffsets threshold comparison sending oneffsets resulted performance improvement full range shifters performance technique becomes exceeds improved encoding technique table speedup original stripes design precision dynamically detected runtime per subgroup activations processed concurrently subgroups belonging group activations processed concurrently finish processing prior advancing next set activations dynamic stripes design improves performance baseline stripes design lags compared pragmatic design skips instead trying exploit precision alone ovel way adjust precision ragmatic pragmatic engine motivated stripes performance depends number activation bits engine opens opportunity value based optimizations performance energy efficiency activation values adjusted fly reduce number bits example determine per layer threshold number significant powers ought kept maintain accuracy eferences judd albericio moshovos stripes deep neural network computing computer architecture letters chen luo liu zhang wang chen sun temam dadiannao supercomputer microarchitecture micro annual international symposium dec albericio judd sharify moshovos bitpragmatic deep neural network computing corr vol xanthopoulos chandrakasan dct core using adaptive bitwidth arithmetic activity exploiting signal correlations quantization ieee journal circuits vol may
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nov connectivity properties factorization posets generated groups henri vivien ripoll abstract consider three notions connectivity interactions partially ordered sets coming reduced factorizations element generated group one form connectivity essentially reflects connectivity poset diagram two bit involved origins algebraic geometry shellability topology propose framework study connectivity properties uniform way main tool certain total order generators compatible chosen element introduction group braid group strands naturally acts elements hurwitz action roughly defined follows ith standard generator acts swapping ith entry entry moreover conjugating one product elements remains unchanged see precise definition action goes back appeared study branched coverings riemann surfaces applies case symmetric group also plays role computation braid monodromy projective curves natural question ask number orbits hurwitz action several results conditions two elements belong hurwitz orbit name deals symmetric group investigates generalized quaternion groups dihedral groups treats dihedral groups dicyclic groups semidihedral groups since hurwitz action preserves multiset conjugacy classes tuple natural study action subset closed gconjugation without loss generality may assume generated monoid authors refer setting equipped group however call structure generated group following fundamental case study hurwitz action factorizations element question number hurwitz orbits mathematics subject classification primary secondary key words phrases generated group braid group hurwitz action factorization poset shellability compatible order well covered poset cycle graph noncrossing partition lattice partially supported public grant overseen french national research agency anr part investissements avenir program reference digiteo project paagt supported austrian science foundation fwf grants latter framework special research program algorithmic enumerative combinatorics henri vivien ripoll let write red set factorizations classical result finite irreducible real reflection group set reflections coxeter element hurwitz action transitive red later generalized finite irreducible complex reflection groups proposition shown recently finite irreducible real reflection group hurwitz action red transitive parabolic element theorem one direction equivalence recently extended affine coxeter groups theorem similar spirit theorem enumerates hurwitz orbits elements alternating group generated also generated groups equipped additional structure partially ordered set construction origin already precisely consider poset prefixes equivalence element ordered containment viewpoint maximal chains correspond factorizations hurwitz action seen method pass one chain another number hurwitz orbits red interpreted connectivity coefficient factorization poset article revolves around relation previously described hurwitzconnectivity two forms connectivity poset motivated graph theory shellability motivated topology main result article following uniform approach proving shellability factorization poset statement result uses two notions formally defined later article namely total order generators compatible definition certain property definition latter property asserts every generator minimal respect given total order find smaller generator common upper cover let fix following notation upcoming three statements let denote group generated monoid assumed closed let denote set generators appear least one factorization theorem red finite factorization poset admits ccompatible order totally well covered respect shellable want emphasize theorem uniformly simultaneously approaches question whether factorization poset hurwitzconnected shellable note far trivial full generality establish factorization poset well covered admits compatible order however special groups framework presented may provide convenient method reach uniform insights connectivity respective factorization posets definition factorization poset well covered respect given total order generators conjecture weaken connectivity properties factorization posets generated groups assumptions theorem bit able prove conjecture part theorem red finite factorization poset admits order hurwitz action transitive red conjecture red finite factorization poset totally admits order shellable main part article deals proper study implications three types connectivity see figure overview also provide three versions conjecture conjectures latter formulated terms particular graph represents local structure factorization poset tool enables prove particular case conjecture see theorem article organized follows section formally define chainconnectivity shellability process recall necessary background define needed concepts section introduce huge class factorization posets arising reflection groups briefly recall definitions state factorization posets possess three connectivity properties section investigate relations three connectivity properties without assumptions heart manuscript section define notion compatible order generators property prove theorem provide equivalent formulation conjecture section culminates proof theorem conclude manuscript section define certain graph essentially recover factorization poset use perspective prove particular case main conjecture three notions connectivity section define three notions connectivity care following three sections serves time preliminary section introduces necessary concepts notions poset terminology section recall basic concepts theory partially ordered sets introduce first notion connectivity partially ordered set poset short set equipped partial order usually write least element greatest element bounded proper part subposet interval set form two elements form covering write equivalently say covered covers let define set coverings consider finite posets chain totally ordered subset meaning henri vivien ripoll fig totally def lem def compatible order totally compatible order conj totally compatible order thm compatible order locally hurwitzconnected shellable reduced cycle graph linear figure implications conjectures several properties omitted arrows coming transitivity whenever occasionally use notation emphasize order elements moreover chain maximal properly contained chain let denote set maximal chains poset graded maximal chains cardinality common cardinality minus one rank denoted first notion connectivity poset springs mind connectivity poset diagram namely graph observe graph trivially connected whenever bounded however poset diagram proper part bounded poset need connected see figure fact interested following stronger version connectivity connectivity properties factorization posets generated groups figure bounded graded poset definition let graded bounded poset define ichain chain graph graph ichain words two maximal chains adjacent chain graph differ exactly one element call connected observe poset diagram proper part poset connected soon rank least three moreover every interval call totally chainconnected easy check induction graded bounded poset totally poset diagram proper part every interval rank connected use characterization following factorization posets generated groups section introduce main construction associates bounded graded poset triple group generated monoid set element fix group subset generates monoid let denote identity call pair generated group define min factorization called let red denote set factorizations order avoid confusion usually write elements red tuples rather words alphabet follows immediately definition satisfies law equality holds say divides case exists factorization prefix areduced factorization gives immediately rise definition following partial order order henri vivien ripoll observe divides lies geodesic right cayley graph words order simply particular acyclic orientation away identity cayley graph respect definition order given perhaps first appeared explicitly case symmetric group notion divisibility goes back next lemma well known experts describes intrinsic recursive structure order proof essentially verbatim proof proposition treats particular case lemma let moreover map order isomorphism particular subinterval interval isomorphic interval form proof pick assumption equations imply follows relations together imply consequence let assume closed case fact subword order red proposition proposition let generating set closed let fix list integers following equivalent exists red aik fix consider principal order ideal generated poset factorization poset whenever clear context omit group generating set observe poset completely determined set maximal chains given setting maximal chains correspond bijectively factorizations let consider map connectivity properties factorization posets generated groups red three following properties whose proofs straightforward used extensively rest paper lemma map bijection inverse given red lemma map lemma preserves map fact closed implies isomorphism type depends conjugacy class lemma closed invariant words moreover gxg note also factorization posets always proposition proposition let closed map restricts interval example let symmetric group permutations well known generated set transpositions since transposition involution generates monoid moreover easy check closed let long cycle factorization poset shown figure reader cordially invited verify lemmas lemma translates case follows replace long cycle map adjusts order letters cycles permutations occurring figure according relative order poset isomorphism proposition verified rotating poset diagram degrees perhaps important consequence assumption closed existence braid group action red thus view lemma also recall braid group strands defined via group presentation henri vivien ripoll figure factorization poset long cycle symmetric group generated transpositions fix define action braid group generator red words generators swap two consecutive factors areduced factorization conjugate one product stays since closed indeed map red straightforward verify action respects relations therefore extends group action red hurwitz action define second notion connectivity used paper definition let define ihurwitz red hurwitz graph graph red ihurwitz view lemma may well define hurwitz graph graph maximal chains point view clearly isomorphic subgraph see definition call hurwitzconnected connected case braid group acts transitively red view lemma sometimes abuse notation write instead figure shows hurwitz graph factorization poset figure shellability posets last notion connectivity important article origins algebraic topology recall set chains graded poset forms simplicial complex order complex denoted consequence hall theorem proposition connectivity properties factorization posets generated groups figure hurwitz graph long cycle symmetric group generated transpositions graded bounded invariant value function least greatest element equals reduced euler characteristic consequently combinatorics provides information topology section want outline simple combinatorial tool edgelabeling may serve learn homotopy type class pure simplicial complexes particularly nice homotopy type shellable simplicial complexes homotopy type fact wedge spheres theorem corresponding homology groups torsionfree ring complexes appendix phrase definition shellability directly terms graded bounded poset transferred pure simplicial complexes via correspondence maximal chains facets definition let graded bounded poset shelling linear order whenever two maximal chains satisfy exists property poset admits shelling shellable observe poset shellable linear order first occurence two successive chains lie different connected components two chains forbid order shelling moreover every bounded poset rank shellable henri vivien ripoll bounded graded poset rank shellable proper part connected bearing mind view shellability sophisticated notion connectivity nice combinatorial way establish shellability exhibiting particular poset map arbitrary partially ordered set naturally extends labeling set maximal chain rising weakly increasing respect partial order falling strictly decreasing chain precedes chain lexicographically smaller respect order every interval exists unique rising maximal chain chain precedes every maximal chain interval poset admits ellabeling proved following fundamental property theorem theorem every poset shellable converse theorem true see instance observe instance lemma case factorization posets coming generated group factorizations induce edgelabeling one main motivations article question whether local criterion guarantee total order turns labeling example consider free abelian group rank isomorphic generated three pairwise commuting elements fix element rst corresponding factorization poset boolean poset shown figure corresponding chain graph shown figure observe graph isomorphic hurwitz graph fix total order reduced factorizations lexicographic order observe unique rising reduced factorization view lemma sequence reduced factorizations corresponds following order straightforward check shelling motivating example let describe one main sources factorization posets arising context generated groups also motivating example present article fix complex vector space consider group unitary transformations element reflection finite order fixes hyperplane pointwise called reflecting hyperplane connectivity properties factorization posets generated groups rst rst rst rst rst rst rst factorization poset prst chain graph factorizathe free abelian group generated tion poset figure figure factorization poset chain graph free abelian group three generators subgroup generated reflections complex reflection group reflection group set reflections generated group easy see closed background reflection groups refer interested reader element said regular eigenvector complement reflecting hyperplanes coxeter element regular element particular order detail refer interested reader instead consequence theorem coxeter elements exist case irreducible fix proper subspace pointwise minimal number reflections needed generate equals dim factorization poset finite irreducible complex reflection group set reflections coxeter element usually called lattice exists vast literature study posets refer reader instance references given therein name comes fact symmetric group set transpositions long cycle poset isomorphism lattice noncrossing set partitions introduced map perhaps first described essentially maps cycles permutation blocks set partition note encountered lattice noncrossing partitions already example next two results show lattices noncrossing partitions possess three types connectivity theorem let finite irreducible complex reflection group let set reflections let coxeter element lattice thus chainconnected theorem let finite irreducible complex reflection group let set reflections let coxeter element lattice henri vivien ripoll shellable particular map defined induces certain total order date however uniform proofs theorems available uniform proof mean proof rely classification complex reflection groups uniform proofs known assumed real reflection group main results respectively remaining complex reflection groups shellability verified case case one main goals work creation uniform framework essentially verify properties means particular total order chosen generating set total case reflection crucial role indeed seen precursor one main definitions article definition interaction different types connectivity section want investigate implications three types connectivity already mentioned following easy observation proposition every bounded graded poset every shellable bounded graded poset proof let bounded graded poset since exists isomorphic subgraph set vertices first claim follows let shellable let indicate shelling assume connected thus minimal index lie different connected components hence since shellable must implies lie connected component consequently lie different connected components thus find lie different connected components contradicts minimality follows connected none converse statements proposition true without assumptions next examples illustrate example converse first statement consider instance finite group given presentation note group isomorphic coxeter group group isometries square presentation given contains dual braid relation see background related groups presentations set clearly closed take ment red implies since rank two however trivially shellable see figure illustration connectivity properties factorization posets generated groups factorization poset prt given presentation hurwitz graph poset figure figure example factorization poset rrrt strr sstr trrr ssst srur susr usrr rrs rrt rrr rss ssru rurr rrv ssus ursr srsu rsur surs urrs srrv rssu vssr rvsr srvs rsrv vsrs rrvr svss rrsv vrss rvrs rsvs rrrt tsss rrts factorization poset prrrt given presentation rtss hurwitz graph poset figure loops drawn figure example factorization poset shellable contains interval rank example next consider infinite group given presentation set closed factorization poset rrrt shown figure corresponding hurwitz graph depicted figure inspection figures see hurwitzconnected shellable since subposet prrr observe example given figure contains interval reason shellable happens exclude situation assume factorization poset totally aware factorization poset totally shellable henri vivien ripoll figure totally poset shellable problem prove disprove following statement finite totally also shellable solution problem would great importance within framework presented could either reduce difficulty prove factorization poset shellable group structure example would exhibit new obstruction shellability follows address problem additional assumptions note arbitrary graded posets examples totally posets shellable consider instance poset figure reproduced page geometric realization dunce hat known theorem however verified every interval hand therefore arise factorization poset generated group seen example factorization poset imply however may add following local criterion make things work definition let factorization poset acts transitively red every call locally theorem let closed fix locally acts transitively red proof let red lemma implies factorizations correspond since connected find sequence moreover say corresponds factorizations view lemma assumption amounts fact since acts transitively red conclude exists connectivity properties factorization posets generated groups rrt rur rrt rsu trr usr rts str urs tss factorization poset prrt prefix order group given presentation sru sus sst hurwitz graph poset figure loops drawn figure example factorization poset interval consequently desired next example illustrates locally actually necessary condition example consider infinite group given presentation set closed figure shows factorization poset rrt corresponding hurwitz graph depicted figure see prrt interval prr compatible section introduce main tool total order compatible concept algebraic generalization compatible reflection order introduced also appeared context reflection groups order make definition work assume red finite equivalent requirement finite definition properties compatible orders let observe trivially red red let total order say factorization red denote rise number factorizations given total order next statement relates rising factorizations hurwitz orbits red specific case henri vivien ripoll proposition let let orb denote number hurwitz orbits red orb min rise total order particular exists total order number hurwitz orbits red equal number factorizations proof hurwitz orbit red assumption finite therefore form total order least one factorizations otherwise would obtain contradiction thus obtain inequality orb rise order remains show equality achieved order since appears exactly one hurwitz orbit red therefore write disjoint union sets form ranges number hurwitz orbits red moreover understand thus define total order setting find rise orb guaranteed find total order rise equals number hurwitz orbits red shown following example example consider group example check computer total order produces least two maximal chains prrrt fact rise rrrt ranges two six however already seen hurwitz graph rrrt connected see figure note example possible find total order every unique factorization observe acts transitively red take instance find red suppose chosen way exactly one elements red four possibilities implies means two factorizations red implies means two factorizations red implies means two factorizations red implies means two factorizations red brings main definition section connectivity properties factorization posets generated groups definition total order exists unique factorization direct consequence proposition get following connection hurwitz action case element length corollary exists order true anymore existence compatible order still implies local see lemma see immediately abelian every total order moreover example illustrates factorization posets admit compatible order however also examples example let continue example let lexicographic order set transpositions following table lists elements length two together sets factorizations highlighted factorizations redt thus conclude hand consider following total order observe two factorizations namely easy way obtain new compatible orders given one lemma let suppose order proof suffices consider case cases follow repeated application immediate need consider length two length two restriction affected cyclic shift indices say element ais ais follows ais henri vivien ripoll unique factorization shift longer shift moreover air shift whenever finally shift shift conclude number factorizations change cyclically shifting order let collect properties orders lemma order restriction proof follows fact element length two lies also lies lemma exists order locally hurwitzconnected proof let restriction corollary implies red unique hurwitz orbit position prove theorem proof theorem suppose admits order lemma implies locally theorem implies acts transitively red however example shows cases red hurwitzconnected exist order remainder section provide evidence orders also closely related shellability factorization poset lemma let order let denote restriction unique factorization minimal maximal respect proof definition lemma exists unique say let amin min amax max proposition implies write amin minimality find amin amin rising factorization uniqueness get amin analogously write amax using hurwitz action rewritten amax maximality find amax implies amax recall total order consider lexicographic order red total order lemma total order lexicographically smallest factorization connectivity properties factorization posets generated groups proof view lemma lexicographically smallest factorization corresponds maximal chain interval view lemma corollary also corresponds maximal chain interval therefore suffices consider case proceed induction claim trivially true suppose claim true let min definition thus write induction assumption lexicographically smallest factorization say factorization satisfies obtain conclude lexicographically smallest factorization clearly consequence lemma find every interval least one maximal chain total order map defined simplicity use notation following natural labeling one motivation paper asking total orders property produces exactly one rising chain per interval words total orders turn ellabeling conjecture case precisely orders conjecture natural labeling respect total order totally ccompatible observe maximal chain rising respect total order corresponding factorization observe one direction conjecture trivially true respect every interval shellable proposition since every interval unique rising chain follows view lemma exists unique factorization element length hence conjecture however suggest exists order order may conclude may exist others though consider instance factorization poset example since rank corollary implies admit order however labeling assigns label sequence one chain label sequence remaining chains clearly observe conjecture requires factorization poset totally drop condition may longer conclude henri vivien ripoll rst wvu tsr interval prefix order quotient free abelian group six generators given relation rst uvw trs str wuv vwu rts srt uwv vuw rst uvw hurwitz graph poset figure figure factorization poset admits compatible generator order neither existence order illustrated following example example let quotient free abelian group six generators given relation rst uvw factorization poset prst shown figure chain graph depicted figure two connected components since generators commute total order always find exactly two rising maximal chains note factorization poset example admits two rising maximal chains every order corresponding chain graph two connected components problem let generated group let element suppose exists order determine connection number factorizations number connected components one quantities always smaller equal fact affirmative solution following problem would enable weaken hypotheses conjecture problem prove disprove following statement finite chainconnected admits order totally property let first state properties factorization posets admitting compatible orders fix total order define words consists upper covers also cover proposition order natural labeling satisfies connectivity properties factorization posets generated groups proof let proceed induction claim trivial assume let length pick must follows minimal lemma implies yields minimal generator lemma implies thus definition total order set empty min factorization posets case empty awarded special name point definition provides different perspective question total orders definition factorization poset well covered respect total order empty min moreover totally well covered respect factorization poset well covered respect appropriate restriction words well covered every atom except smallest one respect find cover smallest atom note also view lemma totally well covered every interval well covered example let continue example fix lexicographic order example list sets transposition since empty min conclude well covered respect example let continue example fix total order observe rst empty even though minimal respect definition prst well covered respect fact well covered respect total order henri vivien ripoll proposition let total order totally well covered respect totally proof suffices prove local variant follows restricting interval repeating argument proceed induction let min first want prove maximal chain component chain running let suppose done otherwise since well covered find let maximal chain stop otherwise repeat construction times reach sequence chains shows connected maximal chain running remains check maximal chains running connected component since totally well covered interval well covered therefore induction hypothesis means two maximal chains connected thus two maximal chains form also connected concludes proof assume fix total order denote total order atoms induced lemma suppose two upper covers proof suppose exists definition find since write since conclude analogously find since minimal lemma implies since obtain construction thus obtain contradiction proceed announced observation totally well covered respect order respect proposition let order suppose totally well covered respect every exists unique factorization proof fix view lemmas suffices consider case proceed induction cases trivial assume connectivity properties factorization posets generated groups lemma implies lexicographically smallest factorization denote factorization let factorization factorization induction hypothesis equal assume let thus write induction obtain lexicographically smallest factorization follows therefore view proposition implies since chosen arbitrarily conclude since totally well covered conclude min yields contradiction theorem let total order totally well covered respect proof definition implication exactly proposition assume observing intervals rank automatically get order moreover since restricts interval restricting accordingly suffices prove well covered pick min let lexicographically smallest factorization restriction since min also know lexicographically smallest factorization since obtain thus consider thus since chosen arbitrarily conclude whenever min precisely says well covered thus obtain following equivalent statement conjecture conjecture totally totally well covered respect order proposition conjectures equivalent proof suppose let order conjecture true would theorem yields well covered thus conjecture would established conversely conjecture true would well covered proposition yields hence conjecture would established remark property modeled condition definition introduces concept recursive atom order bounded henri vivien ripoll graded poset lemma implies order satisfies condition definition consequently conjecture true order totally factorization poset would recursive atom order moreover proofs propositions theorem essentially verbatim parts proof theorem let conclude section proof theorem proof theorem suppose admits order totally proposition implies totally chainconnected moreover theorem implies proposition implies shellable cycle graph section prove particular case conjecture achieve help representation part associate poset simpler structure seen directed labeled graph constructed cycles cycle graph structure retains information rank subintervals allows tackle cases conjectures systematically nice graphical way definition properties cycle graph assume chosen way finite let set length elements set factorizations partitioned hurwitz orbits connected components since finite orbits consists factorizations form pairwise distinct generators represent hurwitz orbit drawing cycle connecting order gives rise following definition definition let factorization poset cycle graph denoted directed labeled graph set vertices oriented edge label given product lemma following properties multiple directed edges loop vertex label set edge labels every element appears label least one edge connectivity properties factorization posets generated groups cycle graph factorization poset figure cycle graph factorization poset figure cycle graph factorization poset figure figure cycle graphs set edges label disjoint union directed cycles corresponding hurwitz orbit red connected component proof immediate definition note also seen union cycles constant label edges figure shows three cycle graphs factorization posets encountered clarity omitted edge labels instead used colors indicate edges label assume admits order follows case lemma every graph unique directed cycle consists edges labeled fact existence order puts bound number edges need remove make acyclic generally directed graph feedback arc set set directed edges whose removal makes acyclic let denote minimal size feedback arc set proposition factorization poset satisfies moreover admits order proof let abbreviate immediate definition needs least number cycles moreover every produces collection cycles say since edges labeled find family cycles obtain suppose implies particular one cycle labeled denote simply let acyclic graph constructed removing edges follows obtained removing exactly one edge since acyclic henri vivien ripoll reduced cycle graph cycle graph figure corresponding order reduced cycle graph cycle graph figure corresponding order figure reduced cycle graphs define following partial order construction linear extension conversely suppose exists order lemma hurwitz action red transitive let write unique hurwitz orbit corresponds unique cycle labeled since order find indices taken mod let remove edge every edge resulting graph satisfies implies cycles left yields let denote acyclic graph constructed last paragraph proof proposition call reduced cycle graph figure shows reduced versions cycle graphs figures cycle graph figure seven different edge labels quickly verified need remove least ten edges make graph acyclic two loops plus one edge eight pairs seen example corresponding factorization admit compatible order agrees proposition proof proposition obtain following corollary corollary let order let dual poset induced linear extension moreover every linear extension lemma order induced linear therefore equal connected directed graph proof follows construction precisely two vertices comparable connected directed path also characterize well covered property inside reduced cycle graph connectivity properties factorization posets generated groups proposition order well covered reduced cycle graph unique sink particular induces linear order well covered proof let minimal element order clearly sink vertex sink exists one vertex equivalent covered corresponding label edge sink exists exactly definition means well covered sink second statement follows naturally since acyclic graph inducing linear order necessarily unique sink following conjecture described terms cycle graph implies conjecture conjecture totally admits order reduced cycle graph induces linear order proof specific case remainder section prove particular case conjecture theorem let finite exists unique conjecture holds let denote unique element theorem denote atom terms cycle graph condition theorem translates follows unique cycle containing vertex counting possible loop cycle labeled particular thanks proposition know factorization starting factor either start explaining strategy proof depends size cycle containing assume following hypotheses conjecture totally exists order particular theorem implies every size cycle containing equal cardinality suppose means commute hypotheses imply written integers conclude cycle graph straightforward obtain one cycle length possibly one two loops reduced cycle graph one oriented edge suppose let write number terms particular suppose either aab first case obtain contradiction second case aab aza forces existence thus contradicting assumption upper covers implies proof henri vivien ripoll theorem reduced two theorems theorem deals case theorem tackles remaining case cases apply following reasoning assume knowledge hurwitz orbit red let appropriate braid group act factorization order exhibit elements step step help proposition deduce size corresponding hurwitz orbits continue find new elements exists may assist process obtained https many cases large enough find exist order remaining cases check explicitly reduced cycle graph induces linear order first need one technical result holds group let define consider sequence given note set finite sequence periodic case size set equals period moreover define alternating order denoted alt minimal alternating products length equal xyx yxy factors factors simply set alt observe alt equality multiple next proposition relates alt sequence proposition let group let define sequence number alt let sequence period alt remark two components equivalence reminiscent two wellknown presentations group associated dihedral group relation xyx yxy corresponds classical braid presentation whereas relations correspond dual braid presentation introduced proof let yields equation hence xck similarly obtain yck denote alternating product xyx consisting factors thus obtain connectivity properties factorization posets generated groups analogously find establishes equivalence proposition following direct consequence proposition let generated group closed let suppose red finite hurwitz orbit red containing size hurwitz orbit red containing proof let hurwitz orbit red containing since red conclude finite write follows sequence proposition period yields alt thus find sequence period well obtain set clearly hurwitz orbit red containing theorem let finite suppose totally exists unique exist order proof assume exists order since totally chainconnected theorem implies particular let write number terms since abb rename thus write appropriate elements continue process finiteness implies find minimal get commute thus obtain contradiction therefore moreover observe let denote cycle abb labeled proposition implies particular graph abb contains also observe exists cycle henri vivien ripoll figure cycle graph abb example corresponding case theorem simplicity loops drawn black even though labels distinct exhibited cycles mutually construction whereas elements length abb conclude feedback arc set abb needs contain two edges label light proposition implies find order abb consequently find order example let illustrate theorem case let abb write obtain factorization poset shown figure corresponding cycle graph shown figure reader invited verify exist order construction however acts transitively red moreover acts transitively red remark indeed observe write proposition implies whole hurwitz orbit forces theorem let finite suppose exists unique conjecture holds connectivity properties factorization posets generated groups figure factorization poset example corresponding cycle graph figure abb factorization poset case theorem cycle graph factorization poset figure simplicity loops drawn black even though labels distinct figure illustration case theorem proof suppose totally exists order particular theorem implies let say abb proposition implies unique possible cycle graph shown figure corresponding factorization poset shown figure four orders namely cyclic shifts see lemma check case reduced cycle graph induces linear order proposition implies unique possible cycle graph shown figure corresponding factorization poset shown figure six orders namely cyclic shifts check case reduced cycle graph induces linear order henri vivien ripoll abbb ccc abb bbb bdd bbe ddd factorization poset case theorem cycle graph factorization poset figure simplicity loops drawn black even though represent distinct elements figure illustration case theorem next let let view proposition implies proposition implies continue find find forces existence find finish iteration obtain since finite find also implies fact cycle graph consists two cycles contradicts assumption unique element covering even without assumption would still obtain contradiction fact find following sequence equalities coming repeated hurwitz moves highlighted words imply contradicts assumption unique element covering observe following cycles well connectivity properties factorization posets generated groups figure cycle graph abbbb corresponding case theorem simplicity loops drawn black even though represent distinct elements thick blue edges represent element affords nontransitive hurwitz action follows feedback arc set needs contain least two edges label view proposition contradicts assumption fact even explicitly find element whose set factorizations act transitively see figure illustration case observe contains interval isomorphic seen previous paragraph restriction interval implies however contradicts assumption references drew armstrong generalized noncrossing partitions combinatorics coxeter groups memoirs american mathematical society christos athanasiadis thomas brady colum watt shellability noncrossing partition lattices proceedings american mathematical society barbara baumeister thomas gobet kieran roberts patrick wegener hurwitz action finite coxeter groups journal group theory tzachi mina teicher graph theoretic method determining hurwitz equivalence symmetric group israel journal mathematics david bessis dual braid monoid annales scientifiques normale david bessis dual braid monoid free group journal algebra david bessis finite complex reflection arrangements annals mathematics philippe biane properties crossings partitions discrete mathematics anders shellable partially ordered sets transactions american mathematical society anders francesco brenti combinatorics coxeter groups springer new york henri vivien ripoll anders adriano garsia richard stanley introduction posets ordered sets springer dordrecht anders michelle wachs lexicographically shellable posets transactions american mathematical society anders michelle wachs shellable nonpure complexes posets transactions american mathematical society thomas brady partial order symmetric group new braid groups advances mathematics thomas brady colum watt partition lattices finite real reflection groups transactions american mathematical society egbert brieskorn automorphic sets braids singularities contemporary mathematics pierre deligne letter eduard looijenga available http xiang dong hou hurwitz equivalence tuples generalized quaternion groups dihedral groups electronic journal combinatorics frank garside braid group groups quarterly journal mathematics masahiro hachimori decompositions simplicial complexes discrete mathematics jia huang joel lewis victor reiner absolute order general linear groups journal london mathematical society james humphreys reflection groups coxeter groups cambridge university press cambridge adolf hurwitz ueber riemann sche mit gegebenen verzweigungspunkten math ann viatcheslav kharlamov viktor kulikov braid monodromy factorizations izvestiya rossiiskoi akademii nauk seriya matematicheskaya germain kreweras sur les partitions non cycle discrete mathematics viktor kulikov factorizations finite groups sbornik mathematics viktor kulikov mina teicher braid monodromy factorizations diffeomorphism types izvestiya rossiiskoi akademii nauk seriya matematicheskaya gustav lehrer tonny springer reflection subquotients unitary reflection groups canadian journal mathematics gustav lehrer donald taylor unitary reflection groups cambridge university press cambridge anatoly libgober invariants plane algebraic curves via representations braid groups inventiones mathematicae henri noncrossing partitions associated complex reflection groups european journal combinatorics henri philippe nadeau alternating group generated lotharingien combinatoire proceedings fpsac victor reiner partitions classical reflection groups discrete mathematics victor reiner vivien ripoll christian stump coxeter elements wellgenerated reflection groups mathematische zeitschrift charmaine sia hurwitz equivalence tuples dihedral groups dicyclic groups semidihedral groups electronic journal combinatorics richard stanley enumerative combinatorics vol cambridge university press cambridge andrew vince michelle wachs shellable poset lexicographically shellable combinatorica james walker poset shellable lexicographically shellable european journal combinatorics connectivity properties factorization posets generated groups patrick wegener hurwitz action affine coxeter groups available institut algebra technische dresden zellescher weg dresden germany address mathematik wien vienna austria address
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topic modeling public repositories scale using names source code vadim markovtsev vadim eiso kant eiso may source madrid spain february languages limited number reserved keywords character based tokens define language specification however programmers rich use natural language within code comments text literals naming entities programmer defined names found source code rich source information build high level understanding project goal paper apply topic modeling names used million repositories perceive inferred topics one problems study occurrence duplicate repositories officially marked forks obscure forks show address using identifiers extracted topic modeling open discussion naming source code elaborate approach remove exact duplicate fuzzy duplicate repositories using locality sensitive hashing model discuss work topic modeling finally present results data analysis together source code tools datasets index open source source code software repositories git github topic modeling artm locality sensitive hashing minhash open dataset ntroduction million public repositories github marked forks makes github largest version control repository hosting service become difficult explore large number projects nearly impossible classify one main sources information exists public repositories code gain deeper understanding software development important understand trends among projects bleeding edge technologies often used first open source projects later employed proprietary solutions become stable exploratory analysis projects help detect trends provide valuable insight industry academia since github appeared movement gained significant momentum historically developers would manually register projects software digests number projects dramatically grew lists became hard update result became fragmented started exclusively specializing narrow notable examples include linux kernel postgresql database engine apache spark framework docker containers technological ecosystems next attempt classify open source projects based manually submitted lists keywords approach works requires careful keywords engineering appear comprehensive thus widely adopted end users practice github introduced repository tags january variant manual keywords submission present paper describes conduct fully automated topic extraction millions public repositories scales linearly overall source code size substantial performance reserve support future growth propose building model names occurring source code applying proven natural language processing algorithms particularly describe weighted minhash algorithm helps filter fuzzy duplicates additive regularized topic model artm efficiently trained result topic modeling open source projects classification reflects drastic variety open source projects reflect multiple features dataset work consists approx million public repositories retrieved github october rest paper organised follows section reviews prior work subject section iii elaborates turn software repositories section describes approach efficient filtering fuzzy repository clones section covers building artm model manually labeled topics section presents achieved topic modeling results section vii lists opened datasets able prepare finally section viii presents conclusion suggests improvements future work elated work academia open source community study presented statistics manually picked topics blincoe studied github ecosystems using reference coupling ghtorrent dataset contained million projects research employs alternative topic modeling method source code million projects instead using ghtorrent dataset prepared open datasets almost public repositories github able comprehensive overview lungi conducted study software ecosystems year github appeared examples work used samples approx repositories proposed discovery methods include natural language processing problem correct analysis forks github discussed kalliamvakou along valuable concerns topic modeling source code applied variety problems reviewed improvement software maintenance defects explanation concept analysis software evolution analysis finding similarities clones clustering source code discovering internal structure summarizing aforementioned works scope research focused individual projects usage topic modeling focused improving software maintenance evaluated software projects concepts extracted using corpus projects giriprasad sridhara yang tan howard considered comments names find semantically similar terms haiduc marcus researched common domain terms appearing source code presented approach papers reveals similar domain terms leverages much significantly larger dataset million repositories bajracharya lopes trained topic model year long usage log koders one major commercial code search engines topic categories suggested bajracharya lopes share little similarity categories described paper since input domain much different industry knowledge companies maintain complete mirror github repositories source focused machine learning top collected source code software heritage strives become open source software sourcegraph processes source code references internal external created complete reference graph projects written golang github centric rather processes dependencies metadata open source packages fetched number repositories analyses dependency graph level projects sourcegraph analyses level functions iii uilding bag words model section follows way convert software repositories model stores project multiset identifiers ignoring order maintaining multiplicity purpose analysis choose use latest version master branch repository treat repository single document improvement research would use entire history repository including unique code found branch preliminary processing first goal process repository identify files contain source code files redundant purpose github machine learning based library named linguist identifies programming language used within file based extension contents modified also identify vendor code automatically generated files first step preprocessing run linguist repository master branch million repositories end million source files high confidence source code written developer project identifying programming language used within file important next step names extraction determines programming language parser extracting names source code highlighting typical task professional text editors ide several open source libraries created tackle task works grammar file written per programming language contains rules pygments high quality communitydriven package python supports programming languages markups according pygments source code tokens classified across following categories comments escapes indentations generic symbols reserved keywords literals operators punctuation names linguist pygments different sets supported languages linguist stores list similar pygments list stored nearly items intersection approximately programming languages programming linguist item type languages common linguist pygments chosen listed appendix research apply pygments million source files extract tokens belong type processing names next step process names according naming conventions example class foobarbaz adds three words bag foo bar baz int wdsize add two wdsize size fig full listing function written python splits identifiers step repository saved sqlite database file contains table programming language name extracted frequency occurance repository total number unique names extracted million fig identifier splitting algorithm python fig name lengths distribution def token token def ret name len name yield yield else part token part continue prev part pos range len part part yield ret part pos pos elif pos yield ret part pos pos elif pos yield ret part pos pos prev last part pos last yield ret last fig influence stemming threshold vocabulary size stemming names common stem names creating nlp since working natural language predominantly english applied snowball stemmer natural language toolkit nltk stemmer applied names characters long research diligent step would compare results without stemming names also predetermine language name available apply stemmers different languages length words stemming applied chosen manual observation shorter identifiers tend collide stemmed longer identifiers need normalized fig represents distribution identifier lengths dataset seen common name length fig plot number unique words dataset depending stemming threshold observe breaking point length vocabulary size linearly grows starting length manually inspected several collisions smaller thresholds came conclusion corresponds best collisions word normalization snowball algorithm chosen based comparative study jivani stems real words acceptable critical minimum since increases number collisions total number unique names million stemming able efficiently data used apache spark running nodes allowed process repositories parallel less day however training topic model one exclude repositories many cases github users include source code existing projects without preserving commit history example common web sites blogs firmwares repositories contain little original changes may introduce frequency noise overall names distribution paper suggests way filter described fuzzy duplicates based bag words model built names source code iltering near duplicate repositories million github repositories october estimation approx million marked forks nearly repositories facto forks marked correspondingly github git commit history colliding hashes repositories may appear user pushes cloned imported repository account without using github web interface initiate fork remove hidden forks initial million repositories still remain repositories highly similar duplicate repository sometimes result git push existing project small number changes example large number repositories linux kernel ports specific devices another case repositories containing web sites created using cloned web engine preserving development history finally large number repositories repositories contain much text content identifiers typically html tags css rules filtering fuzzy forks speeds future training topic model reduces noise obtained every repository naive approach would measure pairwise similarities find cliques first need define similarity weighted minhash suppose two dictionaries mappings unique keys values indicating weights corresponding keys would like introduce similarity measure jaccard similarity dictionaries defined min max weights binary formula equivalent common jaccard similarity definition way minhash algorithm find similar sets linear time weighted minhash algorithm find similar dictionaries linear time weighted minhash introduced ioffe chosen paper efficient allows execution gpus instead large cpu clusters proposed algorithm depends parameter adjusts resulting hash length range sample gamma gamma distribution pdf nif orm compute erk find arg mink return samples thus given supposing integers obtain hash size bytes samples gamma distribution efficiently calculated nif orm uniform distribution developed minhashcuda library python native extension implementation weighted minhash algorithm nvidia gpus using cuda several engineering challenges implementation unfortunately scope paper able hash million repositories hash size equal less minutes using minhashcuda nvidia titan pascal gpu cards locality sensitive hashing calculated hashes dataset perform locality sensitive hashing define several hash tables depends target level false positives elements appear bucket union bucket sets across hash tables specific sample yields similar samples since goal determine sets mutually similar samples consider set intersection instead used implementation weighted minhash lsh datasketch designed corresponding algorithm mining massive datasets lsh takes single parameter target weighted jaccard similarity value threshold minhash lsh puts every repository number separate hash tables depend threshold hash size used default threshold experiments ensures low level dissimilarity within hash table bin algorithm describes fuzzy duplicates detection pipeline step discards less sets aims reducing number false positives bins size distribution step depicted fig clearly seen majority bins size step uses weighted jaccard similarity threshold instead sensitive evident outliers exclusively table reveals different hash sizes influence resulting number fuzzy clones algorithm results approximately sets fuzzy duplicates overall million unique repositories repository appears two sets average examples fuzzy duplicates listed appendix detection algorithm works especially well static web sites share javascript libraries fig lsh hash table bin size distribution fig stemmed names frequency histogram fig fuzzy duplicates detection pipeline fig bag sizes fuzzy duplicates filtering calculate weighted minhash hash size repositories feed hash minhash lsh threshold every repository appears hash tables filter hash table bins single entries every repository intersect bins appears across hash tables cache intersections repository appears existing set create new one filter sets single element resulting number unique repositories corresponds filtered size table every set items calculate precise weighted jaccard similarity value filter less optional return resulting list sets number unique repositories corresponds final size table fig displays bag size distribution raining artm topic model exclusion fuzzy duplicates finish dataset processing pass training topic model total number unique names reduced million million unique names build meaningful dataset names occurrence less excluded final vocabulary chosen frequency histogram shown fig since point exclusion million unique names average size per repository table nfluence eighted ash size number section revises artms describes training topic model performed chosen artm instead topic modeling algorithms since efficient parallel cpu implementation bigartm according benchmarks additive regularized topic model suppose topic probabilistic model collection documents describes occurrence terms document topics fuzzy clones hash size hash tables average bins filtered size final size probability term belong topic probability topic belong document thus whole formula expression total probability accepting hypothesis conditional independence terms belong vocabulary topics taken set simply series indices like solve problem recovering given set documents wnd normally assume nndw number times term occurred document implies terms equally important always true importance means measure negatively correlates overall frequency term let denote recovered probabilities thus problem stochastic matrix decomposition correctly stated ndw stated problem solved applying maximum likelihood estimation ndw max upon conditions value fig artm convergence idea artm naturally introduce regularization one several extra additive members ndw max parameter topics iterations without regularizers iterations regularizers sparsity weight sparsity weight table sed artm meta parameters since simple summation one combine series regularizers objective function example possible increase sparsity make topics less correlated lda model reproduced artm variables effectively calculated using iterative expectation maximization algorithm many ready used artm regularizers already implemented bigartm open source project training vorontsov shows artm trained best regularizers activated sequentially lag relative example first iterations performed without regularizers model reaches target perplexity sparsity regularizers activated model optimizes new members objective function increasing perplexity finally advanced regularizers appended model minimizes corresponding members leaving old ones intact apply sparsity regularizers paper research required leverage others experimented training artm source code identifiers iii observed final perplexity sparsity values change considerably wide range adjustable best training metaparameters given table chose topics merely time intensive label largest amount could label traditional ways determining optimal number topics using elbow curves applicable data consider topics clusters since typical repository corresponds several topics increasing number topics worsens model generalization requires dedicated topics decorrelation regularizer overall number iterations equals convergence plot shown fig achieved quality metric values given table iii average single iteration took minutes complete hardware used bigartm linux environment threads intel xeon cpu computer ram bigartm supports parallel training set number workers peak memory usage approximately possible relax hardware requirements speed training model size reduced set frequency threshold greater value dramatically reduce input data size risk loosing ability model generalize table iii achieved artm metrics metric perplexity sparsity sparsity value trained reference lda topic model provide baseline using lda engine bigartm iterations resulted perplexity sparsity fully dense matrices seen additional regularization made model sparse also yielded better perplexity relate observation fact lda assumes topic distribution sparse dirichlet prior obligatory stand dataset converting repositories topics space let matrix repositories topics space size sparse matrix representing dataset size matrix representing trained topic model size perform matrix multiplication get repository embeddings normalize row matrix metric rtnormed rowwise krt human languages appeared one determine programmer approximate native language looking code thanks stem bias programming languages interesting since information already linguist classification programming languages usually standard library classes functions programs corresponding names revealed topic modeling topics narrow programming language general topics could appear concepts expressive list key words repositories associated unique set names code without special meaning technologies devoted specific potentially narrow technology product often indicates ecosystem community around technology games related video games includes specific gaming engines complete topics list appendix example topic labelled machine learning data science shown appendix observed topics dual need splitted duality sign number topics bigger time topics appear twice need using decorrelation artm regularizer simple reduction increase number topics however solve problems found experimenting topics fig artm topic significance distribution sum along every column matrix indicates significance topic fig shows distribution measure opic modeling results employed topic model unable summarize topics way humans possible interpret topics based significant words based relevant repositories many require manual supervision careful analysis relevant names repositories supervision labour intensive single topic normally takes minutes summarize proper confidence topics required several complete analysis careful analysis sorted labelled topics following groups concepts general broad abstract interesting group includes scientific terms facts world society vii eleased datasets generated several datasets extracted internal github repository storage incorporated recently emerged github data scientists description origin note format definition accessed besides datasets uploaded zenodo doi listed table vii table pen datasets data world name abd doi source code names commits keyword frequencies readme files duplicate repositories description names extracted repositories fuzzy clones excluded considered section iii metadata commits repositories fuzzy clones excluded frequencies programming language keywords reserved tokens across repositories fuzzy clones excluded readme files extracted repositories fuzzy clones excluded fuzzy clones considered section viii onclusion future work topic modeling github repositories important step understanding software development trends open source communities built repository processing pipeline applied million public repositories github using developed open source tool minhashcuda able remove million fuzzy duplicate repositories dataset preprocessed dataset source code names well datasets open presented results reproduced trained artm resulting dataset manually labelled topics data processing model training possible perform using single gpu card moderately sized apache spark cluster topics covered broad range projects repeating dual ones chosen number topics enough general exploration enough complete description dataset future work may involve experimentation clustering repositories topic space comparison clusters based dependency social graphs ppendix parsed languages racket ragel rebol red redcode ruby rust sage salt scala coldfusion common lisp component pascal console coq csharp csound cucumber cuda cython dart delphi dosbatch dylan ecl eiffel elisp elixir elm emacs erlang factor fancy fantom fish fortran foxpro fsharp gap gas genshi gherkin glsl gnuplot golo gosu groovy haskell haxe idl idris igor igorpro inform ioke isabelle jasmin java javascript jsp julia kotlin lasso lassoscript lean lhaskell lhs limbo lisp literate agda literate haskell livescript llvm logos logtalk lsl lua make mako mathematica matlab minid mma modelica monkey standard supercollider swift tcl tcsh thrift typescript vala verilog vhdl vim winbatch xbase xquery xslt xtend zephir ppendix xamples fuzzy duplicate repositories linux kernel abap abl actionscript ada agda ahk alloy antlr apl applescript arduino aspectj autohotkey autoit awk bash batchfile befunge blitzbasic blitzmax bmax boo bplus brainfuck bro bsdmake ceylon cfc cfm chapel chpl cirru clipper clojure cmake cobol coffeescript scheme scilab shell shen smali smalltalk smarty sml sourcepawn splus squeak stan moocode moonscript mupad myghty nasm nemerle nesc newlisp tutorials nimrod nit nix nixos nsis numpy web applications objectpascal ocaml octave ooc opa openedge pan pascal pawn perl php pike web applications plpgsql posh povray powershell progress prolog puppet pyrex python qml robotframework ppendix omplete list labelled topics concepts geometry geometry arithmetic audio bitcoin card games chess hadoop classical mechanics physics color commerce ordering computational physics date time design patterns html parsing email email enumerators mathematical expressions finance trading food pizza cheese beverage calculator genomics geolocalization maps graphs human languages chinese dutch french french german programming languages assembler autoconf clojure coldfusion coldfusion common lisp emacs lisp emulated assembly html human education system java ast bytecode libc php work money employment driving living hexademical numbers general human identifiers identifiers language names advertising facebook javafx engines blocklinear algebra optiers admob mization animation machine learning data antispam php forums science antivirus database acmy cess parsing barcodes browser enparticle physics gines person names charting chat messaging personal information chinese web photography flickr code analysis genplaces transportation eration travel computer memory publishing flask interfaces space solar system console terminal com sun moon cpu kernel trade cryptography trees binary trees date time picker video movies sharding mongodb word term sharding design patterns formal architecture devops drawing portuguese drawing portuguese forms spanish glyphs spanish freetype vietnamese grids tables http auth ibeacons image manipulation lua image processing lua intel simd linear almakefiles gebra mathematics proofs sets operations matlab javascript selectors object pascal jpeg png media players perl metaprogramming python modern frontend python ctypes bower grunt yeoman ruby names starting ruby language networking tensions oauth major web sersql vices string manipulation observer design pattern assembler assembler xpcom online education moodle opengl parsers compilers plotting pointers posix shell vcs promises deferred execution angular proof concept rdf sgml parsing request response requirements dependencies sensors diy devices sockets api sockets networking sorting searching sql database sql xml php projects ssl strings testing mocks text editor threads concurrency typing suggestions dropdowns video player voip web media arch packages web posts web testing crawling web wireless working buffers xml sax xsl xmpp android apps android apache libraries bigdata apache thrift arduino avr technologies chardet python comp vision opencv cordova cpython crumbs cake php curl directdraw directx django web apps cms drupal eclipse swt emacs configs emoji dojo facebook parse sdk ffmpeg fltk fonts fpga verilog freertos embedded glib ionic framework cordova ios networking ios api ios jasmine tests exercises exercism java gui java native interface java web servers javascript ajax javascript dom manipulation joomla jquery jquery grid lex yacc compiler libav ffmpeg linear algebra libraries linux kernel linux wireless lodash mfc desktop applications minecraft mods monads opencl opengl php sites written english people portable document format puppet apps python packaging python scientific stack python scrapers react ros robot operating system ruby rails apps saltstack shockwave flash spreadsheets excel spreadsheets php sqlite stl boost sublime extensions symphony doctrine nlp vim extensions visual basic mssql web scraping winapi wordpress wordpress frontend working pdf php wxwidgets zend framework zlib zlib games graphics unity fantasy creatures repeating topic different key words see section dual topic see section games hello world games minecraft mmorpg pokemon puzzle games rpg fantasy shooters sdl unity engine games web games ppendix words repositories belonging topic achine earning data cience rank word plot numpy plt figur zeros matplotlib dtype fig ylabel xlabel subplot shape pyplot scipy axis arang mean reshap range ylim linspac savefig xlim axes legend bins panda astyp pylab ones xrang len float linewidth linalg norm hist label sum cmap scatter fontsiz self none true xtick figsiz sigma ndarray sqrt rank repository yelp aug python learning tut datajoy eferences sourceforge https ioffe improved consistent sampling weighted minhash sketching proceedings ieee international conference data mining icdm washington usa ieee computer society vorontsov potapenko additive regularization topic models machine learning vol christley madey open source software community structure proceedings north american association computation social organization science naacsos blincoe harrison damian ecosystems github method ecosystem identification using reference coupling proceedings working conference mining software repositories msr piscataway usa ieee press gousios ghtorrent dataset tool suite proceedings working conference mining software repositories msr piscataway usa ieee press lungu reverse engineering software ecosystems phd thesis university lugano sept kalliamvakou gousios blincoe singer german damian promises perils mining github proceedings working conference mining software repositories msr new york usa acm sun liu duan yang exploring topic models software engineering data analysis survey international conference software engineering artificial intelligence networking computing snpd may grant cordy skillicorn using topic models support software maintenance european conference software maintenance reengineering march grant cordy examining relationship topic model similarity software maintenance software evolution week ieee conference software maintenance reengineering reverse engineering feb chen thomas nagappan hassan explaining software defects using topic models proceedings ieee working conference mining software repositories msr piscataway usa ieee press grant cordy skillicorn automated concept location using independent component analysis working conference reverse engineering oct linstead rigor bajracharya lopes baldi mining concepts code probabilistic topic models proceedings international conference automated software engineering ase new york usa acm linstead lopes baldi application latent dirichlet allocation analyzing software evolution seventh international conference machine learning applications dec thomas adams hassan blostein modeling evolution topics source code histories proceedings working conference mining software repositories msr new york usa acm maletic marcus using latent semantic analysis identify similarities source code support program understanding proceedings ieee internationals conference tools artificial intelligence ictai maletic valluri automatic software clustering via latent semantic analysis ieee international conference automated software engineering oct kuhn ducasse semantic clustering identifying topics source code inf softw vol mar thomas mining software repositories using topic models proceedings international conference software engineering icse new york usa acm eddy robinson kraft carver evaluating source code summarization techniques replication expansion international conference program comprehension icpc may mcburney liu mcmillan weninger improving topic model source code summarization proceedings international conference program comprehension icpc new york usa acm saeidi hage khadka jansen itmviz interactive topic modeling source code analysis proceedings ieee international conference program comprehension icpc piscataway usa ieee press sun leung using topic models effectively mining software repositories software maintenance tasks inf softw vol prince nebut dao huchard falleri lafourcade automatic extraction identifier network software international conference program comprehension vol sridhara pollock hill identifying word relations software comparative study semantic similarity tools international conference program comprehension vol yang tan inferring semantically related words software context proceedings ieee working conference mining software repositories msr piscataway usa ieee press howard gupta pollock automatically mining words commentcode mappings proceedings working conference mining software repositories msr piscataway usa ieee press haiduc marcus use domain terms source code international conference program comprehension vol bajracharya lopes mining search topics code search engine usage log proceedings ieee international working conference mining software repositories msr washington usa ieee computer society source https software https https nesbitt http https pocoo pygments generic syntax http porter snowball language stemming algorithms natural language http jivani comparative study stemming algorithms vol ijcta apache https source markovtsev minhashcuda implementation weighted minhash https appeared nickolls buck garland skadron scalable parallel programming cuda queue vol mar zhu https appeared leskovec rajaraman ullman mining massive datasets new york usa cambridge university press blei jordan latent dirichlet allocation mach learn vol mar dellaert expectation maximization algorithm tech georgia institute technology frei apishev shapovalov romov bigartm platform topic https bigartm appeared meaningful collaborative abundant data resource https syed jansen clusters open source ecosystems
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distributed resource allocation using communication applications power networks jan sindri chinwendu enyioha kathryn heal carlo fischione vahid tarokh coordination schemes future power grids require communications since number end devices large bandwidth requirements communication schemes may prohibitive motivated observation study distributed coordination schemes require limited communications particular investigate dual descent distributed optimization algorithm employed power networks using communication iterative algorithm system coordinators broadcast coordinating pricing signals update power consumption based received signal system coordinators update coordinating signals based physical measurement aggregate power usage provide conditions guarantee feasibility aggregated power usage iteration avoid blackout furthermore prove convergence algorithms conditions establish rate convergence illustrate performance algorithms using numerical simulations results show limited communication may viable future smart grids ntroduction network infrastructures central role communication finance technology areas economy instance electricity transmission distribution due massive scale networks distributed optimization algorithms expected increasingly play leading role operation distributed optimization problems commonly solved sub gradient methods primal dual decision variables optimized iteration sort message passing protocol agents network power networks example consumers suppliers communicate power consumption request pricing signals back forth requires ubiquitous communication among consumers suppliers since number end powerconsuming devices large bandwidth requirements schemes may prohibitive however communication infrastructure grid especially power distribution networks still nonavailability communication bandwidth remains challenge given communication bandwidth constraints critical algorithms distributed coordination power work supported chromos project nsf grant nsf career enyioha heal tarokh school engineering applied sciences harvard university cambridge usa email cenyioha kathrynheal nali vahid fischione electrical engineering school access linnaeus center kth royal institute technology stockholm sweden sindrim carlofi networks use significantly less communication overhead instance message passing opposed conventional message passing scheme typically used distributed coordination algorithms said message passing protocol power distribution networks suppliers able send price signal challenging implement especially since widespread system breakdown blackouts occur aggregate power consumption exceeds supply capacity given limits power capacity supply end oneway message passing protocol many power demanding devices network question interest ensure aggregate power consumption users exceed available supply capacity developing distributed algorithms solve problem important algorithm runs users locally compute optimal power allocation aggregate power consumption exceed supply capacity avoid system failure catastrophic blackout work relates closely distributed resource allocation algorithm presented based fast dual gradient method showed dual gradient iteration convergence rate yet primal decision variables converge rate denotes algorithm framework presented however gives guarantee primal feasibility iteration algorithm focus paper developing distributed power allocation algorithm communication power suppliers users used coordinate allocation due critical nature power distribution systems power allocation algorithm presented paper maintains primal feasibility iteration satisfies capacity constraint avoid blackout assumes communication information sharing model supplier network operator sends price signal users users update consumption based signal report consumption back supplier fast convergence rates properties contributions work present distributed dual gradient power allocation algorithm power distribution networks algorithm uses one way communication supplier iteratively broadcasts variable users measures aggregate power usage compute dual gradient since users actually consume power operation route convergence algorithm essential aggregate power usage exceed suppliers capacity avoid blackout show blackouts avoided providing ensure primal iterates remain feasible ever iteration ensure primal feasibility must sacrifice optimal convergence rate gradient methods convex problems lipschitz continuous gradients instead get convergence rate nevertheless prove mild conditions problem structure algorithm attains convergence rate algorithm dual variable still guarantee primal feasibility moreover provide conditions ensure linear convergence rate constant depend number users demonstrate excellent scaling properties algorithm rest paper organized follows following notation definitions introduce system model underlying assumptions distributed solution section sections respectively present convergence analysis convergent rate analysis proposed algorithm section illustration algorithm presented discussed make conclusions section notation definitions definition say function lsmooth gradient continuous definition say function similarly say function concave ystem odel lgorithm paper focus abstract power distribution model simple exposition ignore specific power network constraints consider electric network users whose set denoted single supplier power allocation user denoted total power capacity denoted value power user decided private utility function user would like maximize private mean function user unknown power supplier resource allocation problem power distribution system given following optimization program resource allocation maximize subject assumption convexity function increasing assumption well posed words problem feasible constraint redundant motivated introduction work focus communication protocols solving user problem data private user particular supplier access information consider protocols based following two operations operation communication iteration supplier broadcast users one scalar message referred price power operation feedback information broadcasting price supplier measure deviation total power load supply indicates algorithms solving use operations achieved using duality theory particular achieve solution solving dual problem given follows dual minimize subject dual function dual variables chapter respectively given maximize respectively lower upper bounds power loads user local problem user solve argmax make following assumptions words denotes power demand user price power following relationship lemma strong duality suppose assumptions hold optimal solution optimal solution proof proof follows simply noting ensures satisfies slater condition sufficient zero duality gap convex problems section let demonstrate benefits studying dual problem takes advantage following property general convergence result proposition suppose assumption holds differentiable proof follows directly theorem proposition ensures primal sequence converges however guarantee iterates feasible iteration except limit infeasibilities cause blackouts power network remark therefore essential find conditions ensure feasibility conditions established proof see lemma due proposition solve using dual descent method given iterates gradient given notice exactly feedback provide operation algorithm demonstrate dual descent algorithm implemented supplier users using operations algorithm dual descent using operations local computation users supplier supplier decides initial prize supplier broadcast users user receives user supplier measures supplier updates price remark due communication primal iterates communicated supplier instead users take action aggregate actions measured supplier therefore essential primal problem feasible every iteration algorithm otherwise heightened demand power network result system overload eventual blackout follows study convergence algorithm ensure primal problem feasible every iteration iii onvergence analysis lgorithm section provide rules choosing initial price ensure primal variables feasible throughout algorithm section also prove algorithm convergence rate algorithm certain conditions section following result convergence algorithm standard literature proposition algorithm every limit point solution solution proposition let set optimal solutions assumptions hold algorithm following hold primal variables feasible every iteration convergence rate objective function values moreover optimal convergence rate achieved proof proposition know every limit point moreover since decreasing function equation feasible point also feasible point therefore show holds feasible let show induction assumption let suppose implies convexity implies increasing hence moreover using increasing get rearranging one obtain hence implies hence since increasing must result proved theorem remark general objective functions lipschitz continuous gradients optimal convergence rate using nesterov fast gradient methods see chapter however optimal dual gradient methods ensure feasibility primal problems converging process bringing blackout risk linear convergence rate identify structures ensure convergence rate algorithm start providing linear rate following assumptions utility function local constraints assumption function continuous assumption let set connected interval write terms end points linear convergence rate formally stated follows proposition suppose assumptions hold algorithm converges rate set optimal solutions dist dist optimal convergence rate obtained proof let start showing lemma write convex therefore assumption fact sums convex function lemma get convex since strongly convex one element fact since assumption therefore continuity intermediate value theorem exists show convergence rate algorithm without loss generality suppose proposition choice know hence suppose get inequality follows fact convex implies theorem applying inequality times gives concludes proof note optimal convergence rate depends number users however following mild assumption yields convergence rate dual variable independent number users illustrate excellent scaling properties algorithm assumption let assumption essentially gives user freedom choose utility functions well upper lower bounds constraint utility functions customized reflect preferences priorities user example times day user require power proposition suppose assumptions hold algorithm convergence rate depend number users particular dist dist optimal convergence rate obtained proof proof follows almost steps proof proposition difference dual function instead proposition see lemma appendix next section consider specific form utility function commonly used resource allocation literature particular assume utility function form log parameters may unique different users applying proposition logarithmic utility function assert specific conditions set connected proposition let equation ordered let choose dist dist proof result corollary proposition obtained showing assumption holds see assumption holds note therefore demand estimates time case user user fig using sufficiently small step size feasibility primal problem maintained upper boundary feasible set denoted dotted line implies connected ummary onclusion considered problem allocating power users network one supplier presented distributed algorithm uses communication supplier users coordinate optimal solution showed choose step sizes derive appealing convergence properties dual domain furthermore identified mild modifications power allocation problem results convergence rate algorithm yields convergence rate independent number users network indicate excellent scaling properties algorithm results presented propose paradigm solving distributed resource allocation problems achieve fast convergence rates even information passing mechanism based work interest derive similar convergence rates primal domain addition within context limited communication would like study tradeoffs need primal feasibility convergence rates actual bits transmitted bandwidth using coordination protocol umerical esults section illustrate theoretical findings using two numerical examples examples set use utility function form equation figure depicts first example system consisting two users power suppliers capacity step size observed users assigned socially optimal rate power units primal feasibility satisfied iteration algorithm figure illustrate scenario multiple users initial price cases suppliers capacity assumptions proposition hold case figure depicts aggregate rates allocated users scaled power capacity observed primal variables remain feasible iteration algorithm proved proposition figures respectively illustrate convergence algorithm optimal primal dual variables observed dual variables converge linear rate expected due proposition moreover convergence sensitive number users also agrees results proposition interestingly primal variables also converge linear rate suggest similar convergence results obtained primal variables observed results figures identical explained utilities users identical power supply increases number users however proposition also holds much general assumptions future work consider complicated realistic problems proposition holds ppendix following lemmas used derivations lemma suppose assumption holds set dual function decomposed derivative constant particular proof set pqi defined summing recalling definition dual function get let show theorem holds equality comes using due strong concavity assumption bijective choose primal feasibility convergence primal variable convergence dual variable fig convergence algorithm different number users figures behaviour algorithm identical blue dotted line theoretical bound proposition since concave lemma using get hence holds conclude let show gradient constant result follows combining decreasing lemma function concave proof proof follows directly theorem using convex eferences kelly maulloo tan rate control communication networks shadow prices proportional fairness stability journal operational research society ozdaglar subgradient methods network resource allocation rate analysis information sciences systems ciss annual conference ieee low optimization flow control measurement multiple paths proceedings international teletraffic congress citeseer galli scaglione wang grid grid role power line communications smart grid proceedings ieee vol biglieri coding modulation horrible channel communications magazine ieee vol clarke internet privacy concerns confirm case intervention communications acm vol snow network reliability concurrent challenges innovation competition complexity reliability ieee transactions vol ericsson cyber security power system communicationessential parts smart grid infrastructure power delivery ieee transactions vol gungor sahin kocak ergut buccella cecati hancke survey smart grid potential applications communication requirements industrial informatics ieee transactions vol beck nedic ozdaglar teboulle gradient method network resource allocation problems ieee transactions control network systems vol liu ren yuan wang optimal budget deployment strategy power grid interdiction infocom proceedings ieee april boyd vandenberghe convex optimization new york usa cambridge university press nesterov introductory lectures convex optimization springer yim shin tarokh rate control algorithms fairness guarantees cdma systems wireless communications ieee transactions vol
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ijcsi international journal computer science issues vol issue september issn print issn online cerebellum new computational model reveals primary function calculate multibody dynamics conform formulation lavdim ilir vjosa avni department automation faculty electrical computer engineering university prishtina prishtina kosovo department automation faculty electrical computer engineering university prishtina prishtina kosovo department fundamental engineering subjects faculty electrical computer engineering university prishtina prishtina kosovo department automation faculty electrical computer engineering university prishtina prishtina kosovo abstract cerebellum part brain occupies brain volume contains total number brain neurons new cerebellar function model developed sets cerebellar circuits context multibody dynamics model computations important step controlling balance movement coordination functions performed two oldest parts cerebellum model gives new functional interpretation granule cell circuit including distinct function upper lower golgi cell dendritc trees resolves issue sharing granule cells purkinje cells sets new function basket cells stellate cells according position molecular layer new model enables easily direct integration sensory information vestibular system cutaneous mechanoreceptors balance movement interaction environments model gives explanation purkinje cells convergence nuclei keywords new cerebellar function model cmac cerebellar elementary processing unit golgi cell basket cell stellate cell introduction cerebellum part vertebrate brain part spans sides continually without interruption plays important role lower levels functions motor control relation sensory information coordination complicated bodies balance motor planning according connections parts brain suggested cerebellum involved many functional levels cognition emotional level makes debatable broad diverse functional involvement cerebellum uniform structure neuronal elements present local connectivity circuits parts brain lead authors propose coprocessor function cerebellum based connections neuronal elements cerebellum viewed hierarchically organized system lowest ijcsi international journal computer science issues vol issue september issn print issn online level contains array elementary processing units group neurons single purkinje cell arranged highly ordered fashion starting functions oldest parts cerebellum importance robot dynamics model high performance control algorithms new cerebellar function model developed relates cerebellar circuits multibody multijoint dynamics model computations multibody system dynamics meaning term multibody system broad following term imply set bodies called links assembled together specific way set movement constraining elements called joints interaction bodies happen joints joint concept describes relative position bodies specific construction bodies rigidly connected former case joint infers constraint motion achieved special construction link endings assembled together later case bodies rigidly connected together treated single link minimum number parameters required define relative movement links connected joint denotes joint degrees freedom sum joint degrees freedom gives multibody system degrees freedom joint may actuator intentional altering joint position active joint otherwise joint passive state may change external interactions acting corresponding links links may rigid flexible general links express degree flexibility elasticity normal operating conditions deformation bellow maximal acceptable value theory multibody system used analyze animals skeletal structure vertebrata including humans links represented bones joints usually called articulations muscles used joints actuation typical artificial multibody system sometimes serves synonym robots sometimes structure inspired nature robotics interdisciplinary field many disciplines contribute different aspects development naming mechanics electronics mechatronics control engineering computer science biomechanics etc main topics robotics kinematics dynamics trajectory planning sensing motion control intelligence robot kinematics robot consists links connected joints robot kinematics deals problem finding analytical model describes robot motion respect fixed coordinate frame time without regard forces moments cause motion forward kinematics give position orientation frame attached robot usually frame function joint variables opposite problem finding joint variables given position orientation solved inverse kinematics solution forward kinematics unique whereas solution inverse ijcsi international journal computer science issues vol issue september issn print issn online kinematics necessitating sometime add additional variables sometime use numerical methods robot dynamics general form dynamics equation motion matrix form inverse dynamics robots rigid links derived following formulation gives needed drive joints desired trajectory given joint variables joint position velocity acceleration denotes inertia matrix vector coriolis centripetal terms vector gravity terms friction represented friction model may used usually includes dynamic friction unstructured friction effects possible interaction environment expressed last term vector external forces torques robot jacobian used different forms control strategies feedforward computed torque feedback linearization forward dynamics equation derived solving acceleration used simulation robot movement processing units robot dynamics computation advanced control strategies rely dynamics model robot achieve maintain desired system dynamic response accordance specified performance criteria able track desired trajectories close possible computation appropriate drive joint actuators robot done real time delay caused computer system calculate control algorithm negative influence system performance shorten delay control computer system based single processor computation based formulation dynamics model set forward backward recursive equations computational complexity order number degrees freedom dof robot compared two main additional approaches recursive lagrangian formulation complexity generalized equations motion complexity order classification computational complexity based number multiplications additions dynamics model calculation ndm calculations done single processor computers given execution time per instruction tins total number instructions ndm determine time delay introduced control system computation dynamics model computer architectures many processors recursive computations limit minimum delay proportional number recursion steps nrs even assumption calculation single ijcsi international journal computer science issues vol issue september issn print issn online recursion step one super instruction executes within tins made smaller data dependencies recursion steps case forward backward steps number degrees freedom robot multibody system many slow processors control system must update control values every question control computer architecture set super instructions posses simple solution pops required complex time consuming calculations slow processors store precalculated values table super instruction would one reads input data calculates address input data reads value table writes result output input data directly used address processor would nothing memory sufficient capacity calculation time computer system would despite simplicity approach drawback enormous size memory applications input data space dimension robot dof joint positions velocities accelerations dimension input data space quantized values memory size would indicating exponential growth equal memory size would considered coarse quantization would address byte memory space single byte values stored decreasing memory size unstructured approach sought exploiting structure problem solved dynamics equation motion single joint hkn hknn fkd fks fey gravity term joint written form calculations individual components gravitational constant vector visible explicitly form two advantages first creates gravity terms similar form others second opens possibility compensate directly multibody reference changes orientation relative gravitational field present almost time example humans mobile robots ijcsi international journal computer science issues vol issue september issn print issn online calculate dynamics model single joint seen need small set prototype functions main function prototype form suffice calculation inertial torques gravitational torques gkj torques exerted interaction environment fej calculation coriolis centrifugal torques hkij done using first prototype function calculated defining second prototype function calculating previous terms dynamics equation second prototype done giving constant value equal second variable providing controlled switching mechanism variable calculation dynamic static friction torques fkd fks possible either prototype function mind simplicity compared terms seams like waste computational resources could solved selected prototype function extended second prototype function would become included logical variable switch variable one value two last terms used calculate friction torques computer architecture calculation dynamics model exploits structure model would first layer one processing unit term dof excluding friction terms single super instruction calculation according within tins time second layer one dof summing corresponding results together also within tins time total calculation time would independent dof system first layer architecture resembles single instruction multiple data simd class parallel computer architectures according flynn taxonomy number first layer second layer new structure dimension input space nonlinear function opposed unstructured approach value taken table table memory size single would memory address byte least name tera byte memory space despite enormous reduction value still high different approaches followed reduce value approximation nonlinear functions neural networks one ijcsi international journal computer science issues vol issue september issn print issn online cerebellum cmac small number neuron types present cerebellum common vertebrates arranged highly ordered fashion among largest neuron brain giant purkinje cell numerous type neuron small granule cell organized three layers middle layer called purkinje cell layer contains purkinje cells packed thick layer inner layer purkinje layer granule cell layer inhabits granule cells golgi cells interneurons specific regions outer layer cerebellar cortex named molecular layer consists granule cells axons characteristic form crossing right angle plane purkinje cells dendrite trees present layer also basket stellate cells granule cells excitatory cells cerebellum others inhibitory axons purkinje cell sole outputs cerebellar cortex whereas inputs group mossy fibers climbing fibers lot evidences neuronal connections inside cerebellum higher functional organizations microzones even higher zones considering connections parts brain simple sketch connections cerebellar neurons given fig two main feedforward routes input output first main route afferent mossy fibers project granule cells granule cells project purkinje cell purkinje cell sends efferent axon cerebellar cortex second main route climbing fiber purkinje cell purkinje cell cerebellar cortex connection climbing cell rather specific purkinje cell receives strongest synaptic contact brain exclusively single climbing fiber additional routs basket stellate cells inhibitory interneurons golgi cell forms local feedback loop granule cells additional feedforward route mossy fibers purkinje cell output fiber connectivity like present vertebrates almost invariant fig cerebellar neurons connections signal routes purkinje cell granule cell goc golgi cell stc stellate cell bac basket cell glomeruli granule cell ascending axon parallel fibers adt golgi cell ascending dendritic tree ddt golgi cell descending dendritic tree mossy fibers climbing fiber axon purkinje cell axon ijcsi international journal computer science issues vol issue september issn print issn online difference quantities also additional similar structures vertebrates invertebrates difference anatomical appearance deep folds resulted increase cerebellar cortex purpose compact packing unfolded human cerebellum would approximate dimensions long wide high purkinje cells number purkinje cells rat frog around cerebellum collection numerous elementary processing units epu thousands tens millions may less share processed input data influence epu limited number collaterals output fiber understand function cerebellum first step would understand function single epu neglecting possible mutual couplings would belong higher level organization several epu approach followed renowned theories marr albus formalized later albus cerebellar model articulation controller cmac type artificial neural network ann used robotic controller model retains two main feedforward routs attribution learning mechanism climbing cell route purkinje cell functions perceptron adjustable weights model synapses cell granule cell axons parallel fibers site learning takes place mediated climbing fibers assumed carry error signal model include explicitly molecular layer inhibitory neurons basket stellate cells functionality according theory included possibility negative weights fig shows basic cmac structure granule cell layer processing represented mapping mia input space mossy fibers association space inputs perceptron type ann characterized fast learning attributed local generalization property turn dependent mapping mia selecting proper mappings cmac turn simple perceptron wont able solve even xor problem storage table network properties equivalent radial basis functions rbf anns turn equivalent fuzzy ann quite different ann case multidimensional input spaces standard operation used process receptive fields multiplication operation standard cmac rectangular shaped receptive fields operation used broad range possibilities success network perceptron highly dependent proper selection mapping selected fig cmac structure ijcsi international journal computer science issues vol issue september issn print issn online problem network learn quality quantity learning relies opportunities given network learn associations presented mapping mia since mapping represents processing granular layer turns attention process creating multidimensional receptive fields multiplication really present biological neurons present implemented single neuron group neurons side coordination systems may great benefit dimension input space greatly reduced multiplicative nature speed acceleration dynamics model multiplication two different inputs happen cerebellum possible sites multiplication cerebellum could take place could related dynamics model computation important issue controlling balance assisted evolutionary oldest part cerebellum archicerebellum movement coordination second oldest part cerebellum spinocerebellum known also paleocerebellum new computational model cerebellar function articulated robots dynamics characterized many interactions joints high quality control requires carefully designed controller robot dynamics model usually part controller exception simple cases finding model complicated structures challenge mind oldest parts cerebellum related problems essence contain dynamic interactions better understanding cerebellar function help solve robotic problems ever increasing complexity still number sensors actuators far behind humans animals serve inspiration new model gives new functional description cerebellar circuits revealing relation dynamics calculation model includes new functional interpretation granule cell circuit distinction upper lower dendrite trees explanation scenarios sharing golgi cells purkinje cells new function basket cells new function stellate cells distinct feature upper middle ones definition coding different signals parallel fibers mossy fibers consistency new model direct integration information vestibular system direct integration multilevel interaction environment explanation convergence nuclei cerebellar elementary processing unit cerebellum treated computer architecture several levels organization lowest level contains thousands millions cerebellar elementary processing units cepu one output form axon single purkinje cell functional diagram cepu shown fig numerous inputs come axons granule cells grc know parallel fibers excitatory nature inputs molecular layer inhibitory interneurons stellate cells stc basket cells bac stcs provide input dendrite tree whereas group bacs supply inhibitory synapses structure initial segment soma provide inputs stc bac serve data path shared many pcs ijcsi international journal computer science issues vol issue september issn print issn online receive data many humans million pass data result preprocessing done grcs golgi cells cerebellar input data mossy fibers granule cell circuit granule cell receive inputs number usually distinct generates output single group grc map information input space sparse pgdimensional output space index used indicate existence many different groups marr referred output grc codon representation input albus named mapping process expansion recoding output association input concepts ann output would correspond sort multidimensional basis function sparsity considered important feature new representation influence learning speed capacity fig functional diagram cerebellar elementary processing unit cepu ideal situation mossy fibers provide information form normalized basis functions grc would function multiplier inputs output would normalized higher dimensional basis function ideal approach usually followed using ann goc used deal real ijcsi international journal computer science issues vol issue september issn print issn online situations keep mean output activity controlled level goc influences output two control mechanisms feedback feedforward former acting upper part dendritic tree residing molecular layer synapsing grc axons ascending axon feedforward uses lower dendritic tree synapses according marr output activity controlled influencing grc threshold tree powerfully stimulated albus suggested automatic gain control used maintain constant output activity independent input activity distinction two trees speed acting feedforward path faster acting opinion part cerebellum handling dynamic interactions balance control movement coordination joint decoupling joints multijoint system works mode rate coded version continuous control system functional relation inputs outputs analyzed functional diagram circuit fig see also fig grc receive inputs ljk structures called mossy fiber rosettes cerebellar glomeruli place goc axon terminates influences mossy fiber signals output single grc fig granule cell functional diagram preprocessing input data mossy fibers ijcsi international journal computer science issues vol issue september issn print issn online ggr control done changing threshold ggr control done automatic gain control kth glomeruli specific constants function grc assumed simple summation four incoming signals dendrites followed ggr part may include neuron dynamic nonlinear static characteristics except firing threshold already included plus sign indicates value positive zero neurons nature excitatory inhibitory functionality reversed goc output contrast assumed action potentials dendritic trees simply summed possible behavioral differences taken account functions denote number synapses assumed fixed made upper lower dendritic trees respectively spontaneous firing represented dynamic static without spontaneous firing characteristics goc represented ggo plus sign meaning whereas minus sign preceding indicate inhibitory nature goc static characteristics grc goc given seen goc characterized spontaneous firing activity grc start firing crossing threshold modeled respectively summing active outputs find ggr ggr ggr ggr aim keep small fraction around independence output activity input activity mik achieved contrary conclusion albus positive term ijcsi international journal computer science issues vol issue september issn print issn online function inputs removing making closed loop gain large output active despite output activity kept constant inputs mik form basis functions would result constant first term sparsity controlled widths basis functions could done grc without complicating things cerebellar glomeruli goc conclusion usefulness structure lies interpretation second term since except input information parameters related constructive parameters equation written given equation line negative slope slope effectively turn positive one final action inhibiting inhibitor neuron opinion main functionality behind preprocessing goc cerebellar glomeruli realization linear dependence group outputs additional input simple multiplication operation preparation phase approximating function prototypes given consequence functional interpretation coding used information inputs rate coding goc axon termination corresponding glomeruli outputs provide higher dimensional basis functions space modulated amplitude additional variable outputs used many final function approximation calculating dynamics model terms form set outputs used joints would correspond joint positions vector acceleration removes doubt put marr sharing goc one prediction made interpretation number contact set related dimension joint space dynamic interactions may occur calculate whole dynamics equation term least one goc needed recoding higher dimensional spaces done several goc corresponding subset dimensions number grc dendrites could match dimensionality subspace subsets may realistic meaning like shoulder position head position etc basket cells stellate cells role bac stc according marr setting threshold distinguishing role superficial stc preventing false initial response albus theory assigned function providing negative weights serves function perceptron feature would enable high learning capabilities many models theories cerebellar function include make difference refer together cells concluded stc bac similar types interneurons functional difference synaptic strength bac somatic higher stc dendritic different spatial temporal physiological effects consequences stellate dendritic basket somatic inhibition attributed cerebellum set context balance movement coordination hypothesize different function duty computing dynamic interactions basket cells part circuit calculating coriolis centrifugal terms dynamics equation stellate cells mainly handle calculations friction terms set grc one goc one able approximate ijcsi international journal computer science issues vol issue september issn print issn online calculation product one multidimensional nonlinear function one linearly dependent variable set used calculate terms inertial torques ysk wpcs wpcs calculate coriolis centripetal terms additional multiplication needed one bac axon synapsing soma approximate assumption interaction performs multiplication two signals ijth term joint written hkij kij wpcs kij wpcs nonzero bases functions current joint vector wkijpcs weights pfpc synapses approach would need bac single joint additionally joint speed data needed find suitable solution start group coriolis centrifugal terms hki hki hkin applying given position square basis functions width takes form hki wpcs wpcs kii wpcs kii wpcs bcs kin wpcs kin wbcs kin wbcs last form interpreted sum signals bac controls output one weights wkijbcs represent synaptic connections form input bac uses already present outputs grc need single joint bac needed simplify bac uses basis functions modulated jth joint speed used need resulting total coriolis centrifugal term joint ijcsi international journal computer science issues vol issue september issn print issn online wpcs wpcs wpcs wbcs wbcs bcs wbcs wbcs wbcs wbcs wpcs wbcs wbcs wbcs wbcs value zero kkth term indicates anatomical observations bac inhibit immediately adjacent consistent also physical interpretation interpretation valid wider basis functions justify much less dense dendritic tree bac justification based fact outputs group grc modulated given joint speed carry joint speed information suffice sample sparsely per basis width number inputs bac would meaning ten times less inputs goc necessarily related number synapses even simplification using fixed synapses consequence would bac make synapses ascending axon corresponding grc outputs essentially hard wired connections disagreements functionality grc synapses ascending axon fiber additional possibly adjustable places synapses depending configuration may redundant synapses one adjustable extreme contrast mammals birds total lack bac like fish amphibians reptiles could interpreted lack given type dynamic interactions insignificant contribution different mechanism approximation two friction terms dynamics equation motion dynamic static calculation dynamic friction readily achieved stc would receive inputs sparsely sampled joint speed modulated basis functions like bac output stc would correspond recovered joint speed information stc synapse dendrite would represent dynamic friction coefficient rnd kfd fkd wscs rnd weights wkscs synapses may fixed one whereas adjustable weight one stc wksp friction effect local ideal case one friction term per joint since living organisms many joint many dof driven many paralleled actuators muscles actuator drives given joint need one stc stc superficial regions molecular layer would probably ones related joint deeper one joints dof resultant speed would govern dynamic friction torques calculation static friction term would need grc space encoded joint speed task accomplished smaller goc small number grc outputs grc space coded joint speed information static friction calculated ijcsi international journal computer science issues vol issue september issn print issn online kfs fks wscs case weights wkscs synapses adjustable whereas stc wksp redundant used global adjustment parameter functional integration sensory information types information generated different related balance movement interaction environment easily integrated new functional model cerebellum processed information vestibular systems saccule utricle combined information joints result orientation body relative gravitational field information used computation gravity terms dynamics model similar way done inertial torques difference instead joint accelerations modulatory functions would gravity projections completely way set computation prototypes set grc one goc one used integrate cutaneous mechanoreceptors information movement interaction environment external forces torques places action found processing sensor information dynamic effects computed expressions similar modulatory function six components interaction segment multijoint system last six terms represent group interactions shows one set many possible usually robot interaction environment higher levels organization second level organization cepu done collaterals third level constituted microzones group around arranged narrow longitudinal strip crosses right angle axons terminate group less deep cerebellar nuclei dcn according new model microzone would correspond computational unit computes total joint torques collecting together results individual cepu model help resolve dilemma reason hundreds pcs converge single dcn climbing fibers innervate pcs microzone come group olivary neurons tend coupled gap junctions also axones bac much longer longitudinal direction stay mainly inside microzone thought microzones represent effective cerebellar functional units even higher level organization represents functional modules stripes zones multizonal microcomplexes organizations marked also molecular markers additional fact beside anatomical physiological facts conclusions new computation model cerebellar function tries relate cerebellar neuronal circuits problem multijoint dynamics model computations multiplicative inclusion joint speeds accelerations gravitational acceleration interaction environment make great ijcsi international journal computer science issues vol issue september issn print issn online reduction dimensionality problem space learned extend generalization wide range multiplicative variables model gives functional explanation connections main neuron types consistent model position information bases functions selected width data additionally amplitude modulated rate code additional information source position information mossy fibers form basis functions whereas variables used multiplicatively joint speed joint acceleration use simple rate code compared traditional cmac new model may serve guideline preprocessing possibilities inclusion pure multiplication model biologically plausible solving problems robotics higher level generalization issues resolved context new model functions purkinje cell collaterals collaterals climbing fibers additional information seeking probable explanation branching climbing fiber purkinje cells related redundancy increase resolution maybe load sharing paralleled actuators muscle fibers group related timing issues like interpolation references egidio stefano casali seeking unified framework cerebellar function dysfunction circuit operations cognition frontiers neural circuits januray vol article dan sanes thomas reh william harris development nervous system second edition elsevier academic pess abhinandan jain robot multibody dynamics analysis algorithms springer human body dynamics classical mechanics human movement bruno siciliano lorenzo sciavicco luigi villani giuseppe oriolo robotics modelling planning control springer siciliano bruno khatib oussama eds springer handbook robotics springer gonzalez lee robotics control sensing vision intelligence peter corke robotics vision control fundamental algorithms matlab zhihua darren dawson robust tracking control robot manipulators ieee press rafael kelly victor davila julio antonio perez control robot manipulators joint space springer john hollerbach recursive formulation manipulator dynamics massachusettes institute technology artificial intelligence laboratory memo june revised june lee lee nigam efficient formulation robot arm dynamics control analysis manipulator design university michigan center robotics integrated manufacturing april michael flynn computer architecture pipelined parallel processor design jones bartlett publishers flynn taxonomy last accessed http schuerger liu slater mugnaini unipolar brush cell potential feedforward excitatory interneuron cerebellum neuroscience karl herrup barbara kuemerle ation cerebellum annu rev neurosci ijcsi international journal computer science issues vol issue september issn print issn online ann butler william hodos comparative vertebrate neuroanatomy evolution adaptation second edition john wiley sons curtis bell evolution structures brain behavior evolution gordon shepherd synaptic organization brain fifth edition oxford university press harvey napper quantitative study granule purkinje cells cerebellar cortex rat journal comparative neurology august volume issue pages pellionisz llinas computer model crebellar cortex frog neuroscience vol pergamon press david marr theory cerebellar cortex james albus theory cerebellar function mathematical biosciences february volume numbers pgs james albus new approach manipulator control cerebellar model articulation controller cmac transactions asme journal dynamic systems measurement control september pgs stephen lane david handelman jack gelfand theory development cmac neural networks ieee control systems april wen floiberto ortiz rodriguez marco hierarchical fuzzy cmac nonlinear system modeling ieee transaction fuzzy systems october vol thomas miller iii filson glanz gordon kraft iii cmac associative neural network alternative backpropagation proceedings ieee vol luis masakazu konishi robustnes multiplicative processes auditory spatial tuning journal neuroscience october fabrizio gabbiani holger krapp nicholas hatsopoulos christof koch gilles laurent multiplication stimulus invariance neuron journal physiology paris panagiotis nezis mark van rossum accurate multiplication noisy spiking neurons journal neural engineering christof koch idan segev role single neurons information processing nature neuroscience supplement november vol koch poggio multiplying synapses neurons mckenna davis zornetzer eds single neuron computation zhang wenhui zhu yinfa control space robotic manipulators base neural network ijcsi international journal computer science issues november vol issue yongqiao wei jingdong zhang hou fenglan jia qinglin chang backstepping adaptive fuzzy control robot manipulator ijcsi international journal computer science issues january vol issue maki habib bioinspiration robotics walking climbing robots education publishing egidio cerebellar network revisiting dhe critical issues journal physiology physiol cesana pietrajtis bidoret isope forti granule cell ascending axon excitatory synapses onto golgi cells implement potent feedback circuit cerebellar granular layer jul fabio souza erik schutter robustness effect gap junctions golgi cells cerebellar cortex oscillations neural systems circuits ijcsi international journal computer science issues vol issue september issn print issn online james bower explorations roles molecular layer inhibition regulating purkinje cell responses cerebellar cortex trouble beam hypothesis frontiers cellular neuroscience august volume article jin bao role synaptic plasticity neuronal microcircuit phd thesis biology faculty germany robert sims nicholas hartell differences transmission properties susceptibility depression reveal functional specialization ascending axon parallel fiber synapses purkinje cells journal neuroscience march girija erik schutter james bower ascending granule cell axon important component cerebellar cortical circuitry journal comparative neurology paul dean john porrill ekerot henrik cerebellar microcircuit adaptive filter experimental computational evidence nature reviews neuroscience bishop quantitative analysis distribution purkinje cell axonal collaterals different zones cat cerebellum intracellular hrp study exp brain masao ito cerebellum brain implicit self pearson education richard apps martin garwicz anatomical physiological foundations cerebellar information processing nat rev apr gilad jacobson aspects information processing cerebellar cortex phd thesis hebrew university jerusalem january
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computational capabilities random automata networks reservoir computing david alireza christof portland state university avenue portland usa university new mexico university boulevard northeast albuquerque usa apr dated april paper underscores conjecture intrinsic computation maximal systems edge study relationship dynamics computational capability random boolean networks rbn reservoir computing computational paradigm trained readout layer interprets dynamics excitable component called reservoir perturbed external input reservoir often implemented homogeneous recurrent neural network little investigation properties reservoirs discrete heterogeneous random boolean networks generic heterogeneous dynamical systems use reservoir rbn typically closed system use reservoir extend input layer consequence perturbation rbn necessarily fall attractor computational capability arises separability fading memory inputs find balance properties predictive classification power optimal critical connectivity results relevant construction devices exploit intrinsic dynamics complex heterogeneous systems biomolecular substrates introduction reservoir computing emerging paradigm promotes computing using intrinsic dynamics excitable system called reservoir reservoir acts temporal kernel function projecting input stream higher dimensional space thereby creating features readout layer produce desired output readout layer performs dimensionality reduction traces input signal reservoir two advantages computationally inexpensive training flexibility reservoir implementation latter particularly important systems designed way traditional engineering methods permits computation physical systems show extreme variation interact partially entirely unknown ways allow limited functional control dynamic behavior beyond simple switching makes suitable emerging unconventional computing paradigms computing physical phenomena electronic architectures technological promise harnessing intrinsic computation beyond digital realm enormous potential cheaper faster robust information processing technology maass initially proposed version called liquid state machine lsm model cortical microcircuits independently jaeger introduced variation called echo state network esn alternative recurrent neural network approach control tasks variations lsm esn proposed many different machine learning system control tasks lukosevicius jaeger insofar research focused reservoirs homogeneous transfer functions however due high design variation lack control devices systems heterogeneous connectivity transfer functions since used harness intrinsic computational capabilities physical systems study motivated three fundamental questions heterogeneous reservoirs relationship dynamical properties heterogeneous system computational capability reservoir much reservoir need perturbed adequately distribute input signal may infeasible perturb entire system also perturbation may propagate throughout system due internal topology thus consider size perturbation necessary adequately distribute input signal physical device may difficult observe entire system much system components ought observed extract features input stream model reservoirs random boolean networks rbn chosen due heterogeneity simplicity generality kauffman first introduced model study gene regulatory networks showed boolean networks complex dynamical phase edge chaos average connectivity network hki critical connectivity rohlf showed nearcritical connectivity information propagation boolean networks becomes independent system size packard used evolutionary algorithm evolve cellular automata solving computational tasks found first evidence connects critical dynamics optimal computation detailed analysis mitchell refuted idea accounted genetic drift dynamics evolutionary behavior goudarzi studied adaptive computation task solving boolean networks found learning drives network critical connectivity hkc snyder introduced rbns found optimal task solving networks hki hkc using less restrictive architecture find rbns critical dynamics provided hkc tend offer higher computational capability ordered chaotic dynamics suitable computation reservoir needs eventually forget past perturbations possessing dynamics respond different ways due different input streams first requirement captured fading memory separation property captures second requirement computes distance measurement states two identical reservoirs perturbed two distinct input streams hypothesized computational capabilities optimal separation property highest old input eventually forgotten reservoir occurs fading memory lowest extend measurements described predict computational capability reservoir finite shortterm memory requirement model reservoir computing device made three parts input layer reservoir readout layer fig input layer excites reservoir passing input signal readout layer interprets traces input signal reservoir dynamics compute desired output model reservoir random boolean network rbn fundamental subunit rbn node input connections instant time node assume either two binary states node updates state time according boolean mapping inputs therefore state single node time completely determined inputs time one boolean functions used node rbn collection binary nodes node nodes node receives inputs connected one nodes network model allowed network random two different ways source nodes input chosen nodes network uniform probability boolean function node chosen possibilities uniform probability node sends value output connections destination nodes average connectivity hki study properties rbns characterized nodes average connectivity hki refers instantiations rbns network instantiated collective time evolution time described using xtki xti state node time boolean function governs state update node nodes updated synchronously nodes update state according single global clock signal perspective rbn directed graph vertices hkin directed edges construct graph according random graph model call model heterogeneous rbn node different classical rbn model nodes identical therefore homogeneous original model kauffman assumes static environment therefore include exogenous inputs network use rbns reservoir introduced additional input nodes distribute input signals randomly picked nodes network source nodes links node randomly picked nodes uniform probability input nodes counted calculating hki online computation reservoir extended separate readout layer nodes node readout layer connected node reservoir output node readout layer time yot computed according yot sign parameters weights inputs node reservoir node readout layer common bias readout nodes parameters trained using regression algorithm compute target output paper concerned devices single input node single output node iii measures perturbation spreading rbns typically studied closed systems notion damage spreading used classify rbns dynamics ordered critical chaotic model requires external perturbations must extend notion damage spreading account rbns continuously excited external input since rbn used reservoir closed system propagation external perturbations may behave distinctly propagation damage initial states rbn let rbn nodes average connectivity hki let input stream input layer reservoir hki readout layer found computational capability recurrent neural network reservoirs greatest difference separation fading memory largest coincides critical dynamics therefore want fading memory low separation high define computational capability reservoir input stream length time steps past fig schematic reservoir computing system input layer delivers input signals random nodes inside reservoir readout layer receives output signals random nodes inside reservoir reservoir made collection computing nodes randomly interconnected reservoir creates representation input signals read classified readout layer learning performed training readout layer nodes connections entropy mutual information information theory provides generic framework measuring information transfer noise loss source destination fundamental quantity information theory shannon information defined entropy information source source takes state probability entropy defined variation perturbance spreading input stream variation states rbn driven input streams respectively hamming distance states dynamical system act reservoir needs excited different ways different input streams eventually forgetting past perturbations measurements captured notions separation fading memory however account importance memory reservoir input streams specifically interested separation system time steps past within input stream length ability rbn separate two input streams length differ first time steps given otherwise otherwise amount information contains measure much information transferred source destination calculate mutual information source destination states calculate need calculate joint entropy source destination follows hsd mutual information given hsd see later use entropy mutual information see much information input signals transferred reservoir much information reservoir provide output performing computation order device able generalize reservoir needs eventually forget past perturbations thus define tasks use temporal parity density classification tasks test performance reservoir systems according task system trained continuously evaluate bits injected reservoir beginning time steps past temporal parity task determines bits time steps past odd number values given otherwise train output node form stochastic gradient descent weights incoming connections adjusted every time step training example given system tasks form gradient descent appears yield better training testing accuracies conventional forms use learning rate train weights epochs since dynamics underlying rbn deterministic reset training stream terminate training early accuracy achieved accuracy device stream determined number times output matches expected output specified task divided total number values output stream accuracy input set summed together divided total number input streams set calculate current training accuracy weights output layer trained input streams remain fixed training evaluation every system randomly generate training set testing set stream size training testing sets dependent determined following table temporal density task determines whether odd number bits time steps past values given input stream delay window otherwise input stream delay window fig color online computational capability rbn reservoirs parameters hki hki hki hki hki hki hki computational capability varies according hki left column right column drive reservoirs input streams record number times output device matches expected output generalization capability computed dividing total number times output readout layer matches correct output total number correct outputs process averaged streams general interested finding reservoirs maximize results computational capability computational capability predicted dependent properties reservoir length input stream memory required reservoir properties determined fig color online computational capability rbn reservoirs hki summed calculated dashed curve spline fit highest illustrating connectivity maximizes computational capability particularly high namics due primarily hki number nodes input directly perturbs hki calculate average instantiations figure present results hki produce fig sum dashed curves figs spline fits highlight greatest values figs see rbns critical connectivity hkc tend provide highest high signifies reservoir dynamics ability separate different input streams dynamics determined recent input past input contrast low signifies either following reservoir dynamics frozen separate different input streams effectively traces early perturbations never forgotten reservoir consequence inability compute difficult tasks require long memory consequence great difficulty generalizing past information irrelevant computing correct output readout layer dominates dynamics reservoir output layer unable classify dynamics caused relevant recent input see hki high small due brief afforded rbn subcritical dynamics since network hki little memory computational capabilities unaffected increase demonstrated figs memory early perturbations chaotic reservoirs represented hki characterized sensitivity initial perturbations high separation two identical chaotic systems single bit difference respective input streams eventually become magnified two systems differ states nodes initial perturbation larger differences systems diminish reaching chaotic system could maximize two different ways compute sufficiently short input stream perturb enough system recent input significant effect dynamics past input restriction brief input stream relaxed input stream perturbs nodes possible giving system time propagate perturbations fig hand requires maximizing fig however even distortion staved slowing propagation external perturbations system ultimately fated disorder information optimal perturbation traditional implementations reservoir computing nodes reservoir connected source input signal many task specific generic measures computation reservoirs comprehensively studied however relationship computational properties reservoir number nodes input layer perturbs remains unexplored use information theory characterize computation reservoir information transfer input reservoir reservoir output reservoir computing reservoir dynamical system therefore intrinsic entropy input also calculate entropy order reconstruct desired function output layer pick traces input reservoir dynamics fact reflected entropy change reservoir due input therefore measured using mutual information input reservoir study distribute input reservoir sparsely would thus like find changes function optimal moreover would like know given task solved much information reservoir provide output given desired output reservoir state predictive output equivalent determining much information transferred reservoir desired output show measure using indicates output target order calculate consider instantaneous states reservoir output calculate entropy input need calculate entropy states input take window size example input stream length bits window size time delay input pattern long moving window stream starting time step calculate entropy reservoir consider collection instantaneous reservoir states time step task solving calculate generalization capability rbnrc devices hki tasks random input streams length set parameters instantiate train test devices figs present cubic spline fits average results observe figs critical dynamics provide robust generalization bits bits bits bits bits bits output pattern calculated using output task subtlety arises calculating reservoir entropy since reservoir follows deterministic dynamics input signals perturb reservoir reservoir dynamics identical one repeats experiment reservoirs chaotic dynamics unique mapping reservoir states output patterns therefore reservoir states appear capable reconstructing output completely get correct result one must calculate entropy many streams case since corresponding output patterns change every time mapping reservoir state output appear predictive output feed reservoir randomly chosen input sequences length entropies calculated using states input reservoir desired output takes time interval note need output layer experiments calculations independent training mechanism figure illustrates function reservoirs hki comparison also included ideal reservoir reservoir contains information input indicating reservoir contains required information reconstruct desired output perfectly hki see growing increases maximum level ideal values even nodes reservoirs receiving input systems enough capacity calculate desired output perfectly hki see mutual information increases reaches ideal level systems sparse connectivity input reservoir enough provide required information input reservoir see level reservoir dynamics completely predictive output hki intrinsic dynamics reservoir rich supercritical dynamics mutual information reservoir output reaches peak systems small perturbation quickly spreads reservoir perturbation level enough information reconstruct output fig mutual information input reservoir reservoir output results hki hki hki also included intrinsic information input stream reservoir output cases ideal reservoir hki grows grow maximum level ideal level reservoir systems carry enough information output layer solve task hki grows mutual information grows reaches sufficient level hki mutual information peaks near small perturbation input provide enough information reservoir reconstruct desired output bility task solving ordered chaotic reservoirs evidently solve tasks certain circumstances however ordered networks limited little memory chaotic networks accumulate extraneous information past perturbations demonstrate reduced performance length input stream increases data data data fit fit fit fit fit fit average reservoir indegree hki small little memory processing required reservoir devices reservoir hki achieve perfect generalization figs however ordered networks dominated fading memory hence dynamics retain enough information past perturbations achieve high accuracy increases since dynamics ordered networks determined recent perturbations length input stream irrelevant task solving capability explains generalization ordered networks computing similar seen figs respectively since memory fades quickly ordered reservoir input propagate swiftly network moreover hki network almost certainly possess islands islands unreachable input stream strongly perturb system addition figure demonstrates ordered network hki increases mutual information input reservoir increases fig generalization capability device task dependent hki well task parameters parameters notably chaotic networks achieve maximum generalization capability lower ordered networks ordered networks possess little memory performance drops increases hand chaotic networks perform poorly opposed due inadequately fading memory fit data fit data fit data fit data fit data fit data fit data fit data fit data data data data fit fit data fit data fit data fit fit data fit data fit data fit data data data fig generalization capability device task dependent hki task parameters parameters due high sensitivity initial perturbations generalization capability chaotic networks drop length input stream increases ordered networks possess little memory least robust increase generalization capability better chance see figs increasing tends result higher accuracy respectively therefore increase increase performance reservoir average reservoir indegree hki chaotic reservoirs represented hki dominated separation property result chaotic networks least affected increasing window however high performance possible input stream sufficiently small figs stream length chaotic network high however length input stream increased performance drops significantly even performance networks lower connectivities remain relatively unchanged figs though longer relevant device early perturbations significant effect dynamics makes difficult output layer extract information recent input hand input stream sufficiently short chaotic reservoirs less time distorted early input network chaotic due high connectivity fewer larger connected components less densely connected therefore minimal needed adequately distribute input signal hki network less connectivities explored fig also chaotic reservoir effectively increase computational capability predicted reducing increases time takes perturbations spread system fig evidently chaotic system uses strategy computing seen figure however behavior observed highly parity task speculate due complexity task separation capability significant causes strategy maximizes separation property increasing optimal average reservoir indegree hki observed figure difference separation property fading memory tends maximized connectivity hki evident task solving results increases systems show dramatic drop ordered systems figs simultaneously systems unaffected increase stream length contrast chaotic networks figure observe input signal adequately propagate input signal demonstrated lower small figs however increasing task solving appears afford benefit simply increasing information input stream figs see best critical networks occurs system already achieved maximal input stream reservoir dynamics summary discussion investigated computational capabilities random boolean networks used dynamical component reservoir computing devices found computation tends maximized critical connectivity hkc however reservoir continuously perturbed size perturbations well length time reservoir perturbed must taken account along chaoticity dynamics input stream sufficiently short chaotic systems still perform quite well length input stream increases networks longer differentiate generalize subsets input stream past perturbations may longer relevant computation dominating dynamics hand ordered networks perform well independent length input stream long window computation sufficiently small ordered system retains little information perturbations past network view device also give insight connectivity influences performance reservoir acts input stream set spatiotemporal kernels suitable reservoir needs include diverse set kernels saw connectivity hkc network shows maximal topological diversity dynamics reservoir connectivity hkc therefore act many networks connectivity acting different kernel shown optimal computation occurs recurrent neural networks critical points results provide additional example binary heterogeneous reservoir continuously perturb reservoir underlying rbn model closed system therefore computation dependent attractors must enabled dynamics rbn however circumstances network dynamics fall attractor temporarily indefinitely due frozen dynamics inadequate distribution input signal input stream therefore novel framework explore capacity rbn dynamics information processing rbns studied task solving scenarios goudarzi networks evolve towards criticality although computation still performed attractors study shows unlike findings rbns strong connection computation dynamics optimality computation evidently due critical dynamics network despite differences externally perturbed rbns rbns explored closed system nevertheless observe critical rbns indeed optimal reservoir computing criticality also plays important role biological systems often require optimal balance stability adaptability example shown using compelling theoretical experimental evidence gene regulatory commonly modeled indeed critical conclusion provides intriguing link disparate usages rbn providing evidence critical dynamics desirable heterogeneous substrates findings may relevant development devices exploit intrinsic information processing capabilities heterogeneous physical systems biomolecular nanoscale device networks acknowledgments work supported nsf grants well portland state university maseeh college engineering computer science dergraduate research mentoring program lukosevicius jaeger comput sci rev fernando sojakka ecal springerverlag berlin lnai haselman hauck proc ieee maass markram neural jaeger augustin german national research center information technology technical report gmd kauffman theor biol rohlf gulbahce teuscher phys rev packard dynamic patterns complex systems kelso mandell shlesinger world scientific singapore mitchell crutchfield hraber complex goudarzi teuscher gulbahce rohlf phys rev lett snyder goudarzi teuscher proc thirteenth int conference simulation synthesis living systems alife mit press cambridge bertschinger legenstein advances neural information processing systems edited saul weiss bottou mit press cambridge schrauwen legenstein neural publ math debrecen derrida pomeau europhys lett shannon bell sys tech kesseli theor nyker price aldana ramsey kauffman hood shmulevich proc natl acad sci usa
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lower bounds approximation schemes closest string marek daniel marcin michal sep saket abstract closest string problem one given family strings fixed alphabet task find string minimizes maximum hamming distance string approximation schemes ptases problem known long time acm efficient approximation scheme eptas proposed far paper prove existence eptas closest string fact unlikely would imply fpt highly unexpected collapse hierarchy parameterized complexity classes proof also shows existence ptas closest string running time computable function would contradict exponential time hypothesis institute informatics university warsaw poland cygan supported polish national science centre grant department informatics university bergen norway daniello supported behard grant recruitment programme bergen research foundation institute informatics university warsaw poland supported polish national science centre grant institute informatics university warsaw poland supported polish national science centre grant foundation polish science via start stipend programme work results michal pilipczuk held position warsaw center mathematics computer science institute mathematical sciences india saket department informatics university bergen norway supported parapprox erc starting grant introduction closest string closest substring two computational problems motivated questions molecular biology connected identifying functionally similar regions dna rna sequences well applications coding theory closest string given family strings fixed alphabet length task find one string minimum possible hamming distance number positions different letters consider optimization variant problem said distance minimized decision variant upper bound given input algorithm needs decide whether exists string closest substring general problem strings input family length look string minimizes substring words look fit close possible substring length input strings closest string closest substring well numerous variations problems studied extensively point view approximation algorithms importantly problems classic results providing approximation schemes ptases every possible approximate polynomial time optimum distance within multiplicative factor first ptases problems given running time bounded later improved log andoni sun constitutes current frontier knowledge refer works broad introduction biological applications closest string closest substring related problems well pointers relevant literature one immediate questions stemming works andoni sun whether either closest string closest substring one also give efficient approximation scheme eptas approximation scheme every gives algorithm running time computable function words degree polynomial independent whereas exponential inevitable due happen multiplicative constant standing front running time eptases desirable point view applications since provide approximation algorithms useful practice already relatively small values whereas running times general ptases usually prohibitive general closest substring problem question answered negatively marx using techniques parameterized complexity precisely marx considered various parameterizations closest substring showed parameterized problem remains even binary alphabet means existence algorithm running time total size input would imply fpt highly unexpected collapse parameterized complexity result shows fpt also eptas closest substring excluded indeed eptas existed setting one could time distinguish instances optimum distance value ones optimum distance value thus solving decision variant tractable fpt time using precise results parameterized hardness clique problem marx showed assumption exponential time hypothesis eth states solved time one even expect ptases closest substring running time log computable function refer survey marx examples links parameterized complexity design approximation schemes methodology used marx classic connection parameterized complexity eptases dates back work bazgan cesati trevisan completely breaks applied closest string problem actually admit fpt algorithm parameterized algorithm running time proposed gramm later sun gave algorithm running time efficient alphabets algorithms gramm sun known essentially optimal eth nowadays constitute textbook examples advanced branching techniques parameterized complexity therefore order settle question existence eptas closest string one look substantial refinement currently known techniques approach overcoming issue recently used boucher attribute original idea marx boucher considered problem called consensus patterns variation closest substring goal function total sum hamming distances center string substrings input strings instead maximum among distances problem admits ptas due shown marx tractable parameterized target distance despite latter result boucher managed prove existence eptas consensus patterns would imply fpt main idea provide reduction hard problem clique output target distance bounded function input parameter indeed existence reduction would prove fpt multiplicative gap optimum distances yielded computable function even though output parameter unbounded terms eptas problem could still used distinguish output instances obtained clique fpt time thus proving fpt contribution paper provide negative answer question existence eptas closest string proving following theorem theorem following assertions hold unless fpt eptas closest string binary alphabet unless eth fails ptas closest string binary alphabet running time computable function thus one expect eptas closest string whereas ptases still room improvement running time given sun lower bound theorem worth noting time lower bound approximating closest string also holds general closest substring problem yields significantly stronger lower bound previous log lower bound marx proof theorem follows methodology proposed marx used boucher consensus patterns following theorem main technical contribution work states formally properties reduction theorem integer algorithm given instance clique works time outputs instance closest string alphabet following properties contains clique vertices string contain clique vertices string statement theorem similar core hardness proof boucher however reduction completely different reduction boucher causes computational hardness closest string consensus patterns quite orthogonal consensus patterns difficulty lies picking right substrings input strings substrings known center string easily computed polynomial time since minimizing sum hamming distances closest string substrings pick find center string given input strings computationally hard task minimizing maximum hamming distances center rather sum theorem follows immediately combining theorem known parameterized hardness results clique gathered following theorem setting theorem theorem corollary following assertions hold unless fpt clique solved time computable function unless eth fails clique solved time computable function main idea proof theorem encode vertices given graph almost orthogonal family strings log strings used identifiers vertices fact almost orthogonal means identifiers two distinct vertices differ approximately positions hand log whole space strings embedded size polynomial using properties reduction promised theorem designed careful construction notation log denote logarithm positive integer denote length string denoted alphabet two equallength strings hamming distance denoted number positions different letters binary alphabet hamming weight string denoted number complement string binary alphabet denoted obtained replacing vice versa note selection gadget rest paper fix following constants since divisible integers first prove among binary strings logarithmic length one find family almost orthogonal strings balanced hamming weight proof simple greedy argument lemma exist positive integers divisible following property let integer let denote exists set following properties distinct moreover given constructed time polynomial proof let denote binary entropy log log suppose positive integer divisible well known integers lemma let denote follows suppose positive integers divisible log log since choose integer divisible choose large enough integers hence verify choice satisfies required properties consider following greedy procedure performed start strings marked unused consecutive rounds perform following pick yet marked used add mark every used clear step procedure constructed family satisfies properties hence suffices prove procedure performed least rounds note number strings marked used round hand set strings hamming weight exactly infer means algorithm able find unmarked least rounds hence construct family easy implement algorithm polynomial time using fact size polynomial adopt constants given lemma notation let also fix let set strings given lemma shall call selection strings define set forbidden strings follows words comprises strings almost diametrically opposite string following lemma asserts properties shall need later lemma suppose following assertions hold exists proof property follows directly definition proceed proof suppose could take suppose means exists equivalently hand also construct taking set positions size letters flipping letters positions replacing vice versa set positions always exists implies claim suppose otherwise since also must exist equivalently hence triangle inequality infer contradiction assumption implied indeed hence definition implies thus satisfies required properties main construction section provide proof theorem let input instance clique let let constants given lemma assume otherwise instance solved constant time let run algorithm given lemma computes set selection strings let set forbidden strings defined section note computed polynomial time directly definition due present construction output instance closest string set partition set positions strings length blocks blocks length special balancing block length contiguous subset positions denote substring formed positions let first discuss intuition choice solution string makes consecutive blocks encode selection vertices vertices mapped strings family constraint strings consist two subfamilies ssel sadj following roles strings ssel ensure block solution picks substring close element selection element encodes choice ith vertex strings sadj verify vertices chosen pairwise different adjacent hence form clique small technical caveat strings ssel sadj intended hamming distance solution string slightly different role balancing block equalize distance simple additional construction proceed formal description since let arbitrary bijection family ssel consists strings function binary string length string constructed follows block put string block put string consisting zeroes string consisting ones balancing block put string thus also ssel constructed time directly definition family sadj consists strings ordered pair vertices either equal function string constructed follows block put string block put string block put string consisting zeroes string consisting ones balancing block put string consisting zeroes thus sadj constructed time directly definition set ssel sadj concludes construction correctness verified two lemmas mirror properties listed theorem lemma contains clique vertices exists string proof let construct putting block zeroes positions balancing block first take string ssel since lemma infer since due regardless value finally obviously hence second take string sadj since different adjacent whereas either equal without loss generality suppose second case symmetric due property lemma obviously finally every hence regardless value strings match positions summarizing lemma string contains clique vertices proof first prove block close selecting element claim exists unique proof uniqueness follows directly property lemma triangle inequality suffices prove existence let sake contradiction suppose lemma infer exists let take defined follows majority positions contain one otherwise also define consider string ssel follows consequently contradiction assumption let claim vertices different adjacent proof sake contradiction suppose either equal define follows majority positions contain one otherwise constructed string sadj observe since similarly consequently contradiction assumption claim asserts indeed lemmas conclude proof theorem taken constant larger conclusions paper proved closest string eptas assumption fpt moreover stronger assumption exponential time hypothesis one also exclude ptases running time computable function however fastest currently known approximation scheme closest string running time leaves significant gap known upper lower bounds despite efforts unable close gap hence leave open problem references andoni indyk optimality dimensionality reduction method annual ieee symposium foundations computer science focs october berkeley california usa proceedings pages ieee computer society bazgan approximation phd thesis paris sud french boucher lokshtanov consensus patterns probably eptas algorithms esa annual european symposium patras greece september proceedings volume lecture notes computer science pages springer full version available http cesati trevisan efficiency polynomial time approximation schemes inf process cygan fomin kowalik lokshtanov marx pilipczuk pilipczuk saurabh parameterized algorithms springer flum grohe parameterized complexity theory texts theoretical computer science eatcs series berlin gramm niedermeier rossmanith algorithms closest string related problems algorithmica wang finding similar regions many sequences comput syst wang closest string substring problems acm lokshtanov marx saurabh slightly superexponential parameterized problems proceedings annual symposium discrete algorithms soda san francisco california usa january pages siam sun efficient algorithms closest string substring problems siam marx closest substring problems small distances siam marx parameterized complexity approximation algorithms comput
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nov automorphism groups quandles related groups valeriy bardakov timur nasybullov mahender singh abstract paper study different questions concerning automorphisms quandles conjugation quandle conj group determine several subgroups aut find necessary sufficient conditions subgroups coincide whole group aut particular prove aut conj aut either one groups solves problem big list takasaki quandles abelian group prove group inner automorphisms inn coxeter group extends result theorem describes inn aut abelian group without study automorphisms certain extensions quandles determine interesting subgroups automorphism groups quandles also classify finite quandles action aut introduction quandle algebraic system whose axioms derived reidemeister moves oriented link diagrams algebraic system firstly introduced joyce name quandle matveev name distributive groupoid invariant knots precisely oriented diagram oriented knot one associate quandle change apply reidemeister moves diagram moreover joyce matveev proved two knot quandles isomorphic weak equivalent exists homeomorphism probably orientation reversing maps years quandles investigated various authors order construct new invariants knots links see example knot quandle strong invariant knots however usually difficult understand two knot quandles isomorphic sometimes homomorphisms knot quandles simpler quandles provide useful information helps understand two knot quandles isomorphic example fox useful invariants links representations knot quandle dihedral quandle elements leads necessity studying quandles necessarily knot quandles algebraic point view quandles turned useful different branches algebra topology geometry since connections several different topics permutation groups quasigroups symmetric spaces hopf algebras paper investigate different questions concerning automorphisms quandles also study connections groups automorphisms quandles groups automorphisms mathematics subject classification primary secondary key words phrases quandle automorphism quandle braid group enveloping group coxeter group valeriy bardakov timur nasybullov mahender singh groups closely related quandles point approach purely algebraic make reference knots however would interesting explore implications results knot theory paper organized follows section recall definitions notions used paper give several examples quandles section recall notion enveloping group quandle prove properties group particular study relations automorphism groups proposition section study automorphism group conjugation quandle conj group find groups imbedded aut particular prove aut conj aut either one groups theorem section study automorphism groups core quandles particular takasaki quandles section classify quandles automorphism group aut acts theorem section study automorphisms certain extensions quandles determine interesting subgroups automorphism groups quandles theorem section compare group inner automorphisms group automorphisms quandles give simple solution analogue burnside question quandles proposition finally section introduce new ways constructing quandles groups quandles propositions acknowledgement authors greatful jente bosmans vrije universiteit brussel several remarks suggestions section results given sections supported russian science foundation project bardakov dstrsf project singh results presented sections supported research foundation flanders fwo grant nasybullov strongly grateful definitions notations examples quandle algebraic system one binary operation satisfies following three axioms map bijection simplest example quandle trivial quandle set quandle lot interesting examples quandles come groups let group elements denote conjugate arbitrary integer set operation forms quandle quandle called conjugation quandle group denoted conjn sake simplicity use symbol conj denote conjugation quandle note quandle trivial define another operation group namely set operation also forms quandle quandle called core quandle group denoted core particular abelian group quandle core called automorphism groups quandles related groups takasaki quandle abelian group denoted quandles studied takasaki cylcic group order takasaki quandle called dihedral quandle elements denoted quandle operation given rule mod automorphism abelian group set operation forms quandle quandle called alexander quandle abelian group respect automorphism denote alex takasaki quandle particular case alexander quandle alex alexander quandles studied example subset quandle called center quandle particular conj center quandle coincides center group sets conjn coincide bijection called automorphism quandle every elements equality holds particular trivial quandle group aut automorphisms quandle consists permutations elements aut full permutation group elements second third axioms quandle follows map automorphism quandle subgroup hsx group aut called group inner automorphisms quandle denoted inn every automorphism inn called inner automorphism quandle element subset inn called orbit element denoted orb set orbits elements quandle denoted orb quandle called connected set orb one element otherwise called disconnected dihedral quandle connected odd conjugation quandle conjn nontrivial group always disconnected enveloping groups quandles quandle denote group set generators set relations group called enveloping group quandle example trivial quandle group free abelian group rank enveloping group dihedral quandle elements interesting generators defined following relations using first relation remove element list generators obtain following relations first second relations together imply equality belongs center analogically first third relation imply belongs center therefore looking square first relation relation equivalent relation proved following valeriy bardakov timur nasybullov mahender singh proposition enveloping group dihedral quandle two generators two relations particular homomorphic image braid group strands subgroup braid group called pure braid group strands group normal following short exact sequence groups defining homomorphism rule following short exact sequence groups center extension split lemma proved enveloping group connected quandle decomposed result emplies following proposition gives split decomposition group proposition enveloping group product proof using proposition direct calculations easy check general case following proposition enveloping groups proposition let quandle finitely many orbits abelianization enveloping group isomorphic finite derived subgroup enveloping group finitely generated proof let generated elements defined relations index uniquely determined multiplication modulo relation form isomorphism commutator subgroup generated commutators number commutators finite dihedral quandle elements connected orbit therefore proposition abelianization enveloping group isomorphic result follows proposition since enveloping group dihedral quandle presentation natural map maps element corresponding generator induces homomorphism quandles conj homomorphism necessarily injective example section joyce noticed quandle elements operation map maps element therefore injective following result shows relation automorphism group automorphism group case natural map injective automorphism groups quandles related groups proposition let quandle natural map maps element correspoding generator injective autq denotes subgroup aut consisting automorphisms keep invariant natural map autq aut given isomorphism groups proof map obviously bijection autq moreover indeed map autq aut obviously homomorphism need prove bijection aut since generated set injective map surjectivity let aut composition natural inclusion extended group homomorphism free group elements induces group homomorphism easy see automorphism keeps ivariant inverse unfortunately quite difficult check natural map injective however proposition group aut subgroup group aut subgroup necessarily normal aut indeed trivial quandle natural map injective enveloping group free abelian group rank group aut isomorphic consists possible permutations elements group aut general linear group gln acts naturally contains least elements denote map belongs group aut gln map belongs group aut gln since automorphism form belong aut therefore aut normal subgroup aut automorphisms conjugation quandles let group automorphism since every automorphism induces automorphism conjn therefore aut aut conjn course group aut necessarily normal aut conjn example cyclic group prime order order aut equal quandle valeriy bardakov timur nasybullov mahender singh conj trivial aut conj proper normal subgroup alternating group order therefore aut normal aut conj show aut aut conj lemma let group conj automorphism exists element proof xyz side xyz since arbitrary element group element must belong corollary group trivial center aut aut conj proposition group nontrivial center aut aut conj proof let denotes bijection conj fixes elements except let show aut let elements since hence automorphism since map belong aut corollary let group aut aut conj proposition proved group exists embedding groups aut aut conj going prove group aut also imbedded aut conj proposition let group group aut conj contains subgroup isomorphic direct product aut proof consider map aut aut conj maps every automorphism group map conj conj acts conj following rule since aut map maps maps therefore bijection conj easy show preserves operation conj hence automorphism conj automorphisms aut element hence therefore homomorphism aut aut conj particular automorphism groups quandles related groups therefore hence injective homomorphism aut aut conj consider map aut conj maps every permutation elements map acts conj following rule similarly case map easy see injective homomorphism aut conj let subgroup aut conj generated aut groups obviously normal moreover normal aut conj therefore aut corollary let group group aut conj contains subgroup isomorphic group proof group acts faithfully right multiplications therefore subgroup abelian conj trivial quandle aut conj isomorphic abelian proposotion group subgroup aut conj following result says finite abelian group conclusion proposition correct proposition let finite abelian group aut conj contains subgroup isomorphic aut proof aut aut conj aut conversely since abelian group quandle conj trivial quandle aut conj full permutation group order group aut equal group aut order therefore following result describes finite groups aut conj aut theorem let finite group aut conj aut proof corollary proposition imply aut conj aut need prove groups trivial center groups aut conj aut suppose according proposition group aut consists automorphisms form automorphism permutation elements following element denote automorphism quandle conj form valeriy bardakov timur nasybullov mahender singh since element map must form exist two elements equality axy hand therefore contradicts choice proved elements product belongs means abelian therefore proposition already mentioned proposition proved group aut conj contains subgroup isomorphic semidirect product aut following theorem describes finite groups satisfy equality aut conj aut gives answer problem theorem let finite group aut conj aut either one groups proof theorem simple corollary center group trivial aut conj aut aut groups following equalities aut conj aut aut conj aut aut conj aut need prove groups without center groups groups aut conj aut element denote automorphism quandle conj form group generated isomorphic subgroup aut conj isomorphic aut consists automorphisms form aut see proposition details need prove every automorphism conj form aut either one groups suppose permutation elements denote automorphism conj form suppose aut exist two elements equalities imply since choose belong aut therefore every two elements product automorphism groups quandles related groups belongs means therefore abelian conj since every automorphism fixes unit element since aut conj equality every permutation nontrivial elements automorphism particular nontrivial elements order suppose exist least distinct elements since every permutation elements fixes automorphism map extended automorphism means contradicts fact elements distinct therefore order less equal since nontrivial elements order one groups automorphisms core quandles takasaki quandles following proposition analogue proposition core quandles groups proposition let group center aut core subgroup isomorphic group aut proof denote operation quandle core automorphism therefore induces automorphism core denote subgroup core generated automorphisms induced automorphisms aut element denote bijection core form obviously automorphism core denote subgroup core generated since every automorphism intersection trivial automorphism automorphism therefore normal subgroup group generated hence semidirect product aut abelian group core quandle core takasaki quandle proposition strong modification proved theorem theorem additive abelian group without aut aut inn obvious elementary abelian takasaki quandle trivial quandle elements group automorphisms quandle isomorphic group inner automorphisms trivial attempt complement theorem considering finite abelian groups let mij symmetric matrix entries mii mij coxeter group type group following presentation mij valeriy bardakov timur nasybullov mahender singh results coxeter groups found example theorem let finite additive abelian group cyclic decomposition znr znk divides even number znr largest odd order component odd order component group inn isomorphic coxeter group mij symmetric matrix mij proof definition inn hsx easy see idt note exponent elements using direct calculations therefore group inn generated involutions subject relations idt order determine distinct generators inn notice let two elements thus number distinct involutions inn equal znr largest odd order component odd order components proves inn isomorphic mij matrix mij arbitrary group discription group inner automorphisms inn core quotient subgroup group presented proposition theorem theorem give alternative description inn core case abelian group cyclic group order takasaki quandle dihedral quandle theorem implies following result corollary let integer group inner automorphisms inn dihedral quandle isomorphic coxeter group mij matrix mij corollary group inner automorphisms inn elementary abelian group proof case corollary group aut isomorphic semidirect product acts permuting components automorphism groups quandles related groups proof quandle elements corollary group inn isomorphic group let following automorphism easy see inner automorphism order moreover therefore group subgroup aut symmetric group write since aut aut remark dihedral quandle known disconnected two orbits define map rule direct calculations show map automorphism therefore arguments proof proposition work case quandle observation also shows bijection quandle permutes orbits quandle neccessarily automorphism quandles action automorphism group let group acts sex left exists homomorphism maps element permutation set number say acts pair distinct elements exists element gxi sake simplicity say acts independently quandle group inner automorphisms inn acts naturally group inn acts say quandle particular quandle connected mccarron proved finite quandle least four elements proposition moreover dihedral quandle elements quandle therefore higher order transitivity exist quandles least four elements investigate following problem problem integer classify finite quandles aut acts lemma let quandle automorphism valeriy bardakov timur nasybullov mahender singh proof therefore conversely therefore theorem let finite quandle following statements equivalent either trivial quandle aut aut acts proof implications obvious thing need prove quandle action aut either trivial isomorphic dihedral quandle since aut acts therefore lemma quandle trivial statement proved element exists element since therefore three distinct elements let arbitrary element since aut acts exists automorphism therefore three elements since aut acts exists automorphism therefore since aut acts exists automorphism therefore repeating argument possible triples following relation dihedral quandle corollary aut acts acts following question remains open question classify finite quandles action aut automorphisms extensions quandles let quandle set map called constant quandle cocycle satisfies two conditions ids examples constant quandle cocycles found example set constant quandle cocycles denoted quandle set constant quandle cocycle consider set operation given rule automorphism groups quandles related groups set operation quandle called extension respect denoted quandles introduced exists surjective quandle homomorphism onto namely projection onto first component hand set trivial subquandle two constant quandle cocycles said cohomologous exists map relation cohomologous equivalence relation equivalence class constant quandle cocycle called cohomology class denoted symbol denotes set cohomology classes constant quandle cocycles following result generalizes result lemma formulated alexander quandles lemma let quandle set constant quandle cocycles cohomologous quandles isomorphic proof denote operation quandle operation quandle since constant quandle cocycles cohomologous exists map denote map maps pair pair map obviously bijection isomorphism quandles lemma let quandle set constant quandle cocycle aut map defined rule gives left action aut induces action proof using direct calculation easy check constant quandle cocycle map left action aut cohomologous constant cocycles exists map acting equality map map given therefore cohomologous map maps class class well defined map group acts set left set forms subgroup called stabilizer lemma group aut acts valeriy bardakov timur nasybullov mahender singh set constant quandle cocycles following result gives relation groups aut aut theorem let quandle set constant quandle cocycle exists embedding aut aut proof automorphism aut permutation belongs aut define map rule map obviously bijection since aut definition map equality elements automorphism injective map aut aut maps pair automorphism aut thus hence homomorphism remark element aut map defined automorphism reversing preceeding calculations easy see aut map defined automorphism abelian group denote map maps element map following call function quandle map constant quandle cocycle two quandle said cohomologous corresponding constant quandle cocycles homologous set cohomology classes quandle second cohomology group quandle coefficients group given quandle denote symbol quandle formula operation form quandle called abelian extension plenty quandles abelian extensions smaller quandles see details results faithfull extensions quandles also correct abelian extensions quandles particular theorem emplies following result corollary let quandle abelian group quandle embedding aut aut aut automorphism groups quandles related groups automorphisms recall automorphism group called element exists element since conjugation element defines inner automorphism group automorphism every element exists inner automorphism every inner automorphism obviously burnside formulated following question true automorphism group inner burnside answered question negatively constructing example finite group automorphism inner quandles formulate two different definitions automorphism automorphism quandle called automorphism strong sense every element exists element automorphism quandle called automorphism weak sense every element exists inner automorphism inn every automorphism strong sense obviously weak sense however general case opposite correct since group inner automorphisms inn hsx coincide set conjugation quandle conj group two definitions automorphism nilpotent group two definitions automorphism equivalent also quandles conjn denote qinn set weak sense automorphisms quandle set qinn obviously subgroup aut contains group inner automorphisms inn trivial quandle groups inn qinn trivial following question analogue burnside problem quandles question exist quandle inn qinn group automorphism constructed burnside quandle conj strong sense automorphism induced automorphism gives negative answer question going construct simpler example quandle conjugation quandle possesses quasiinner automorphism lemma connected quandle aut qinn proof let automorphism since connected element exists inner automorphism inn therefore aut qinn proposition odd inn qinn proof theorem odd group aut isomorphic group multiplicative group ring also theorem odd group inn isomorphic group therefore odd groups inn aut coincide since odd dihedral quandle connected lemma valeriy bardakov timur nasybullov mahender singh qinn aut inn exists automorphism inner remark proposition group aut isomorphic inn automorphism direct calculations show automorphism therefore inn qinn would interesting explore whether similar result holds constructions new quandles using automorphisms section describe general approaches constructing quandles using automorphism groups group map aut said compatible every element following diagram commutes aut aut denotes inner automorphism aut form aut example map idg map maps every element inner automorphism induced compatible proposition let group aut compatible map set operation quandle moreover injective map satisfies yxy proof need check three axioms quandle equality given condition proposition therefore first quandle axiom faithful element map automorphism therefore bijection second quandle axiom faithful since compatible map elements therefore third quandle axiom faithful set operation quandle compatible map satisfies equality therefore injective compatible map aut form idg quandle constructed proposition trivial compatible map aut maps element inner automorphism quandle constructed proposition quandle following proposition gives another interesting construction quandle automorphism groups quandles related groups proposition let two quandles let aut aut two homomorphisms set operation quandle following conditions hold proof need check three quandle axioms element equal either equal therefore first quandle axiom faithful element map acts map map therefore map bijection analogically map bijection second quandle axiom faithful let elements elements belong third quandle axiom obviously works elements since works let case using equality equalities third quandle axiom faithfull similarly using equality obtain third quandle axiom trivial maps required conditions proposition faithful quandles always define quandle structure another example let involutary quandle quandle equality holds let quandles isomorphic isomorphisms using direct calculations easy check maps satisfy conditions proposition therefore define quandle structure every involutary quandle particular every core quandle core group quandle mentioned section quandle map injective obtained two trivial quandles using procedure proposition homomorphisms maps homomorphism maps automorphism permute valeriy bardakov timur nasybullov mahender singh references brenti combinatorics coxeter groups graduate texts mathematics springer new york andruskiewitsch grana racks pointed hopf algebras adv bardakov dey singh automorphism groups quandles arising groups monatsh doi burnside outer isomorphisms group proc london math carter survey quandle ideas introductory lectures knot theory ser knots everything world sci hackensack carter elhamdadi nikiforou saito extensions quandles cocycle knot invariants knot theory ramifications carter elhamdadi saito twisted quandle homology theory cocycle knot invariants algebr geom topol carter kamada saito diagrammatic computations quandles cocycle knot invariants diagrammatic morphisms applications san francisco contemp amer math providence clark elhamdadi saito yeatman quandle colorings knots applications knot theory ramifications clark saito algebraic properties quandle extensions values cocycle knot invariants knot theory ramifications fox quick trip knot theory topology garcia iglesias vendramin explicit description second cohomology group quandle math doi gorin lin algebraic equations continuous coefficients problems algebraic theory braids math hulpke connected quandles transitive groups pure appl algebra joyce classifying invariant knots knot quandle pure appl algebra kamada knot invariants derived quandles racks invariants knots kyoto geom topol geom topol coventry mccarron small homogeneous quandles issac international symposium symbolic algebraic computation acm new york matveev distributive groupoids knot theory russian mat nelson combinatorial revolution knot theory notices amer math nosaka central extensions groups adjoint groups quandles arxiv przytycki elementary invariants knots banach center publications knot theory guide quasigroups latin quandles quasigroups related systems takasaki abstraction symmetric transformation tohoku math tamaru homogeneous quandles prime cardinality math soc japan sobolev institute mathematics acad koptyug avenue novosibirsk russia novosibirsk state university pirogova novosibirsk russia address bardakov automorphism groups quandles related groups katholieke universiteit leuven kulak sabbelaan kortrijk belgium address department mathematical sciences indian institute science education research iiser mohali sector nagar manauli punjab india address mahender
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bouveyron latouche mattei exact dimensionality selection bayesian pca charles bouveyron laboratoire umr cnrs paris descartes sorbonne paris mar pierre latouche laboratoire samm paris mattei laboratoire umr cnrs paris descartes sorbonne paris abstract present bayesian model selection approach estimate intrinsic dimensionality dataset end introduce novel formulation probabilisitic principal component analysis model based prior distribution context exhibit expression marginal likelihood allows infer optimal number components also propose heuristic based expected shape marginal likelihood curve order choose hyperparameters nonasymptotic frameworks show simulated data exact dimensionality selection approach competitive bayesian frequentist methods keywords dimensionality reduction marginal likelihood multivariate analysis model selection principal components introduction computer age characterized surge multivariate data often difficult explore describe natural way deal datasets reduce dimensionality interpretable way trying loose much information accordingly wide range techniques developed years principal component analysis pca perhaps earliest techniques remains today one widely used jolliffe cadima introduced pearson rediscovered hotelling beginning twentieth century pca indeed ubiquitous role statistical analysis since introduction electronic computation recent exemples include climate research hannachi expression studies massive text mining zhang ghaoui even deep learning chan exhaustive overview past applications pca defer reader monograph jolliffe recent review paper jolliffe cadima specifically pca consists simple procedure practitioner orthogonally projects multivariate data space spanned eigenvectors associated largest eigenvalues empirical covariance matrix dimension representation learnt way simply number eigenvectors called principal components pcs kept projection however may come surprise spite popularity method authoritative solution widely accepted choosing many exact dimensionality selection bayesian pca pcs computed common practice choose dimension considering eigenvalues scree sample covariance matrix technique popularized cattell largely modified perfected last fifty years jackson zhu ghodsi often chosen pca used building block within larger algorithmic framework see bouveyron example cluster analysis evangelopoulos latent semantic analysis however refined approaches also developed earlier works based hypothesis testing jolliffe section suggested wold developed years bro known effective wide variety settings josse husson another fruitful line work follows seminal article tipping bishop recast pca simple inferential problem model called probabilistic pca ppca led several methods dimensionality selection frequentist ulfarsson solo bouveyron passemier bayesian bishop minka hoyle perspectives aforementioned methods based asymptotic considerations however recently proven asymptotic framework hard thresholding eigenvalues surprisingly suffices provide optimal dimensionality gavish donoho thus path efficient schemes finding number pcs goes study criteria overlooked past natural answer provided exact bayesian model selection previously used price computationally expensive markov chain monte carlo mcmc sampling hoff present prior structure based ppca model allows exhibit expression marginal likelihood leading efficient algorithm selects number pcs without asymptotic assumption specifically rely normal prior distribution loading matrix gamma prior distribution noise variance imposing simple constraint hyperparameters respective distributions show allows data marginally follow generalized laplace distribution leading efficient computation marginal likelihood also propose heuristic based expected shape marginal likelihood curve order choose hyperparameters simulated data demonstrate approach competitive compared methods especially non asymptotic settings less observations variables setting core many practical problems genomics chemometrics section briefly review ppca present several dimensionality selection techniques based model new prior presented section together derivation expression marginal likelihood heuristic choose hyperparameters also presented numerical experiments provided section choosing intrinsic dimension probabilistic pca let assume centered independent identically distributed sample observed aim projecting onto subspace retaining much variance possible observations stored matrix bouveyron latouche mattei probabilistic pca ppca model assumes observation driven following generative model wyi gaussian latent vector parameter matrix called loading matrix gaussian noise term model instance factor analysis first introduced lawley tipping bishop presented thorough study model particular expanding result theobald proved generative model indeed equivalent pca sense principal components retrieved using maximum likelihood estimator wml specifically matrix ordered principal eigenvectors diagonal matrix corresponding eigenvalues wml arbitrary orthogonal matrix sound probabilistic framework dimension selection recast model selection problem standard techniques available review important ones next subsection model selection ppca problem finding appropriate dimension seen choosing best model within family models first popular approach would use likelihood penalization leading choice log pen pen penalty grows methods include popular akaike information criterion aic akaike bayesian information criterion bic schwarz refined approaches bai however merits mainly asymptotic main interest paper investigate scenarios penalty term usually necessary avoid selecting largest model constrained ppca model called isotropic ppca bouveyron proved regular maximum likelihood suprinsingly consistent theoretical optimality method also asymptotic fact directly maximizes likelihood criterion derived based asymptotic considerations makes particular interest within scope paper another interesting set techniques essence bayesian model selection kass raftery bic actually approximates marginal likelihood case ppca violated regularity conditions drton plummer refined approach followed minka derived laplace approximation marginal likelihood technique albeit asymptotic proven empirically efficient several scenarios exact dimensionality selection bayesian pca another interesting framework considered literature case grow infinity several consistent estimators proposed penalization point view bai passemier using stein unbiased risk estimator ulfarsson solo bayesian context hoyle scenarios growing importance fall beyond scope paper focused setting potentially fewer observations variables automatic dimension selection methods available exact dimensionality selection ppca prior section present prior structure leads expression marginal likelihood ppca model consider regular ppca model already defined wyi parameter matrix rely gaussian prior distribution loading matrix gamma prior distribution noise variance specifically use gamma prior gamma hyperparameters together gaussian priors wjk precision hyperparameter within framework bayesian model uncertainty kass raftery posterior probabilities models written marginal likelihood data note expression also involves model prior probabilities paper simply consider uniform prior computing integral marginal likelihood usually comes price various approximations bishop minka hoyle expensive sampling hoff however specific choice priors imposing constraint respective hyperparameters obtain expression marginal likelihood bouveyron latouche mattei theorem let prior logmarginal likelihood model given log log log log log log log modified bessel function second kind order detailed proof theorem given next subsection best knowledge result first computation marginal likelihood ppca model worth mentioning slightly different context ando also derived marginal likelihood factor analysis model student factors similarly bouveyron derived exact marginal likelihood noiseless ppca model order obtain sparse pcs gaussian priors loading matrix extensively used past bishop archambeau bach bouveyron worth noticing use gamma prior variance parameter rather peculiar indeed bayesian hierarchical models choose priors variances choice often motivated conjugacy properties see george mcculloch linear regression example murphy wider setting derivation provided next subsection notably explains gamma prior actually arises naturally derivation marginal likelihood begin shortly reviewing generalized laplace distribution prove key within ppca framework distribution introduced kotz detailed overview see kozubowski definition random variable said multivariate generalized asymetric laplace distribution parameters characteristic function generalized laplace distribution elliptically contoured referred symmetric generalized laplace distribution elementary properties generalized laplace distribution discussed kozubowski list ones consider proof theorem proposition galp cov moreover positive definite density given det exact dimensionality selection bayesian pca proposition let galp galp independant random variables galp proposition directed consequence expression characteristic function generalized laplace distribution another appealing property multivariate generalized laplace distribution interpreted infinite scale mixture gaussians gamma mixing distribution property called transmutation ding blitzstein proposition generalized transmutation let gamma independent galp proof result see kotz chap prove theorem first study marginal distribution signal term following mattei state following lemma lemma let random matrix columns following distribution gaussian vector independent obtain galp proof let column prove lemma demonstrate follow gal distribution use decomposition let since standard gaussian follows distribution equivalently gamma distribution therefore gamma moreover note sign since sign independent sign therefore according proposition galp since following galp distribution use proposition conclude galp focus second term involving noise vector bouveyron latouche mattei lemma let gamma galp proof transmutation considered indeed noise written therefore transmutation allows conclude proved signal noise term follow marginally generalized laplace distribution use proposition ensures assuming sum two generalized laplace random vectors generalized laplace random vector galp using expression density generalized laplace distribution eventually end expression marginal likelihood theorem choosing hyperparameters obtain expression marginal likelihood shown sufficient assume two hyperparameters remain henceforth tuned shape parameter gamma prior precision hyperparameter developed heuristics purpose first observation grows expected decay signal part model expressive prior information distilled model roughly centering gamma priors estimates precisely heuristic choose order control diffusiveness loading matrix variance specifically made choice experiments chose estimator mean smallest eigenvalues covariance matrix see tipping bishop complex estimates may considered passemier regarding remaining parameter propose heuristic based following statements made regarding problem dimension selection overestimation preferred underestimation since loosing information much damageable representation parsimonious enough consequently marginal likelihood curve function dimension two distinct phases first one signal dimensions added true value second one noise dimensions added thus built simple heuristic criterion judge relevance choice shape marginal likelihood curve first slope first part curve maximum lower slope second part means choice leads underestimation therefore discarded second criterion equal discrete second derivative marginal likelihood curve evaluated maximum exact dimensionality selection bayesian pca dimensionality dimensionality dimensionality dimensionality dimensionality figure different shapes marginal likelihood curve growing values corresponds maximum heuristic criterion describe subsection true dimensionality order select hyperparameter leading strong distinction two phases criterion eventually maximized grid values scheme hyperparameter choice illustrated fig using simpler simulation scheme described subsection numerical experiments section perform numerical experiments order highlight main features proposed approach compare methods simulation scheme assess performance algorithm referred hereafter ngppca short consider following simulation scheme following experiments follow simulation setup proposed bouveyron based isotropic ppca model therefore simulate data sets following isotropic ppca model assumes covariance matrix two different eigenvalues instead ppca model case ratio snr hereafter simply defined snr simulation set control strength signa varies explore different ratios orthonormal matrix drawn uniformly random data eventually generated according centered gaussian distribution covariance matrix times times diag finally number variables fixed experiments number observations varies range bouveyron latouche mattei introductory examples first conduct two small simulations illustrate behavior algorithm difference laplace approximation minka consider two scenario simple case harder realistic one simple scenario consider setup snr simple scenario first illustrate heuristic hyperparameter tuning displaying marginal likelihood curves different values fig heuristic criterion allows find desired shape leading correct dimensionality estimation gif animation displaying values criterion large grid values provided online fig compare results algorithm laplace approximation marginal likelihood minka case methods recover true dimensionality data confident choice posterior probability true model higher approaches two curves similar shape compliance expected shape detailed subsection challenging scenario consider setup snr gif animation illustrating hyperparameter tuning provided results compared laplace approximation fig regarding exact approach left panel marginal likelihood curve extremely similar shape one first simulation shape satisfactory maximum heuristic criterion actually corresponds true dimensionality although also finds correct dimensionality laplace approximation wrongfully prefers simpler models precisely top two models chosen laplace approximation posterior probability posterior probability contrast algorithm favors posterior probability posterior probability prefering overestimation underestimation exact method appears less likely destroy valuable information would damaging dimensionality selection context summary experiments confirm expected behaviors laplace approximation first scenario asymptotic assumption laplace approximation much relevant second setup method rely assumption much less impacted reduction sample size benchmark comparison dimension selection methods section focuses comparison methodology dimension selection methods consider possible scenarios snr grid going repetitions made case compare performance technique based prior following four competitors laplace approximation minka benchmark bayesian method dimension selection http http laplace exact prior exact dimensionality selection bayesian pca dimensionality dimensionality methodology laplace minka figure exact ngppca left laplace approximation minka right simpler simulation scenario curves desirable properties detailed subsection find correct dimensionality laplace exact prior dimensionality dimensionality methodology laplace minka figure exact ngppca left laplace approximation minka right challenging simulation scenario methods select correct dimensionality laplace approximation prefers underestimation satisfactory behavior exact computation gives much acceptable curve bouveyron latouche mattei fraction correct answers gcv laplace ratio fraction correct answers gcv laplace ratio fraction correct answers gcv laplace ratio fraction correct answers gcv laplace ratio figure percentage correctly estimated dimensions different sample sizes averaged replications case ngppca competitors different ratios top bottom data sample sizes respectively exact dimensionality selection bayesian pca generalized approximation gcv josse husson known give state art results many scenarios see vast simulation study josse husson profile likelihood approach zhu ghodsi represents screebased techniques popular several different contexts fogel evangelopoulos approach bouveyron maximizes criterion likelihood notice specifically adapted simulation scheme advantage allow consider technique simulated data performance metric chose percentage correct answers given algorithm standard measure used simulations studies see minka hoyle ulfarsson solo results presented fig one first notice methods vastly outperform profile likelihood approach seems small sample sizes second generalized crossvalidation gives often satisfactory results fails competitive methods laplace snr high approach good behavior especially small partly explained fact designed simulation setup finally approach consistently outperforms bayesian method laplace approximation method gives satisfactory results settings high low snr moderate small conclusion pca descriptive exploratory tool model therefore unique dimension selection method uniquely preferred sometimes relevant information may actually lie within last pcs jolliffe section however pca ubiquity statistical world makes necessary search guidance procedures help practitioner choose number pcs need even critical data scarce particularly expensive work deviating usually adopted asymptotic settings step direction regarding future work exact computation model posterior probabilities may used perform bayesian model averaging hoeting predictive contexts potential applications could involve principal component regression jolliffe chap image denoising deledalle deep learning chan concluding note work comes illustration exactly computing marginal likelihood sometimes easier expected although recent asymptotic approximations drton plummer mcmc arsenal friel wyse deal marginal likelihoods argue like lin finding exact expressions important task deemed untractable hastily bouveyron latouche mattei acknowledgement part work conducted pam visiting university college dublin funded fondation sciences paris fsmp references akaike new look statistical model identification ieee transactions automatic control ando bayesian factor analysis factors exact marginal likelihood journal multivariate analysis archambeau bach sparse probabilistic projections advances neural information processing systems pages bai determining number factors approximate factor models econometrica bishop bayesian pca advances neural information processing systems pages bouveyron girard schmid data clustering computational statistics data analysis bouveyron celeux girard intrinsic dimension estimation maximum likelihood isotropic probabilistic pca pattern recognition letters bouveyron latouche mattei bayesian variable selection globally sparse probabilistic pca technical report bro kjeldahl smilde kiers component models critical look current methods analytical bioanalytical chemistry cattell scree test number factors multivariate behavioral research chan jia gao zeng pcanet simple deep learning baseline image classification ieee transactions image processing deledalle salmon dalalyan image denoising patch based pca local versus global proceedings british machine vision conference pages ding blitzstein representation transmutation arxiv preprint exact dimensionality selection bayesian pca drton plummer bayesian information criterion singular models journal royal statistical society series statistical methodology mar issn evangelopoulos zhang prybutok latent semantic analysis five methodological recommendations european journal information systems fogel young hawkins ledirac inferential 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application text data advances neural information processing systems pages exact dimensionality selection bayesian pca zhu ghodsi automatic dimensionality selection scree plot via use profile likelihood computational statistics data analysis
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cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems please note provided author accepted manuscript version may differ final published version feb cite publication please use final published version personal use material permitted permission author copyright holder must obtained uses current future media including reprinting republishing material advertising promotional purposes date submission june date acceptance may please contact provide details believe document breaches copyrights remove access work immediately investigate claim distributed control prescribed performance synchronization unknown nonlinear networked systems sami hashim frank lewis paper proposes distributive cooperative tracking control prescribed performance function ppf highly nonlinear multiagent systems ppf allows error tracking predefined large set trapped predefined small set key idea transform constrained system unconstrained one transformation output error agents dynamics assumed completely unknown controller developed strongly connected structured network proposed controller allows agents follow trajectory leader node satisfying necessary dynamic requirements proposed approach guarantees uniform ultimate boundedness transformed error adaptive neural network weights simulations include two examples validate robustness smoothness proposed controller highly nonlinear heterogeneous networked system time varying uncertain parameters external disturbances index performance transformed error distributed adaptive control consensus transient error introduction coordination animals social groups ants foraging birds swarming fish schooling forth attracted attention researchers develop distributed collaborative control engineering applications especially robotic systems collaborative work agents empower group execute hard large tasks whcih individual agent simply achieve instance capable simplifying mission dividing several complementary tasks exchanging vital information may save group increasing productivity certain cases imitating biological behaviors combining collaborative framework autonomous vehicles could beneficial many applications surveillance inspection space explorations many areas agents sami department systems engineering king fahd university petroleum minerals dhahran selferik corresponding author hashim department electrical computer engineering university western ontario ontario canada frank lewis research institute university texas arlington texas email lewis manuscript received april revised required work distributed manner communicate using particular network configuration hence agents often called nodes customized follow least one leaders network formed nodes called communication graph vertices graph comprise nodes edges represent connections nodes graph may undirected case difference two vertices associated edge hand may directed one vertex another indicates direction information flow nodes respective neighbors literature one pioneer work addressed consensus systems consensus passive nonlinear systems studied node consensus cooperative tracking problem investigated many research work cooperative tracking control studied single node high order dynamics robust adaptive control addressed solve consensus problem multiagent systems connecting undirected graph type network topology problem addressed case network directed graph communication topology also unlike control solve problem case leader possesses state trajectory developed distributed control heterogeneous agents unknown nonlinear dynamics authors considered nodes single integrator later high order systems addressed although previous studies assumed input function node dynamics known however cooperative tracking control problems systems unknown input function studied fuzzy proposed approximate unknown nonlinear dynamics input functions centers output membership functions determined based trials one common ground shared studies considered unknown nonlinear dynamics input function linear parameters lip see typical instances intermittent control ultimate stability tracking error main concern studies cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems model uncertainties external disturbances affect considerably closed loop performance controlled agents particular error proven ultimately bounded converge within residual set size depends unknown bounded terms representing model uncertainties external disturbances prediction transient error behavior becomes almost impossible establish analytically prescribed performance simply means tracking error confined within arbitrarily small residual set convergence error within given range addition convergence rate less prescribed value less prescribed constant prescribed performance robust adaptive control mainly developed provide smooth control input signal soft tracking address problem accurate computation upper bounds systematic convergence summary adaptive prescribed control ensures convergence error within prescribed bound overshot less prescribed value uniform ultimate boundedness property transformed output error smooth adaptive tracking developing cooperative adaptive control systems based prescribed performance yield many merits prescribed performance means secluding system tracking output error start within large set convert systematically arbitrarily small set predefined range error prescribed performance convergence rate less predefined value maximum overshoot undershoot exceed certain limit prescribed performance cooperative adaptive control ability increase robustness control reduce control effort proper computations upper lower bounds prescribed performance functions provide soft tracking error systematic convergence robust adaptive control ppf proposed develop controllers feedback linearizable systems since many papers considering prescribed performance used linearly parametrized neural network approximate unknown nonlinearities disturbances see instance adaptive control developed particular system structures systems affine systems high order nonlinear systems papers made various assumptions regarding continuity input matrix using trial error methods defining neural weights adaptive control prescribed performance proposed mimo uncertain chaotic systems using model reference adaptive control almost studies considered single autonomous systems paper propose decentralized neuroadaptive cooperative control fleet autonomous systems linked communication digraph different synchronization errors nodes expected dynamic responses satisfy predefined characteristics determined designer dynamics node subject model uncertainties unknown bounded external disturbances control scheme based prescribed performance transient well dynamic performance node synchronization error knowledge never treated paper original prescribed performance scheme proposed modified take considerations interactions nodes created consensus algorithm presence random disturbances time instant hence stable dynamics bounded smooth decentralized control inputs guaranteed rest paper organized follows section preliminaries graph theory prescribed performance bounds presented problem formulation local error synchronizations sense ppf presented section iii section develop control law formulation prove stability connected digraph section includes simulation results verify robustness proposed control finally conclude suggest future directions research section notations following notations used throughout paper absolute value real number euclidean norm vector frobenius norm matrix trace matrix set singular values matrix maximum singular value minimum singular value indicates matrix positive definite positive identity matrix unity vector required appropriate dimension preliminaries basic graph theory graph denoted nonempty finite set nodes vertices set edges arcs edge node node topology weighted graph often described adjacency matrix aij weights aij otherwise aij throughout paper topology fixed selfconnectivity element aii graph directed undirected directed graph called diagraph weighted node pni defined sum row aij define diagonal matrix diag graph laplacian matrix set neighbors node node neighbor node node get information cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems node necessarily vice versa undirected graph neighborhood mutual relation direct path node node sequence successive edges form diagraph spanning tree node called root directed path root every node graph diagraph strongly connected ordered pair nodes directed path node node iii problem formulation consider following nonlinear dynamics ith node state node control signal node unknown disturbance node unknown nonlinear dynamics assumed lipschitz assumed agent state vector results include kronecker products global graph dynamics described leader state defined desired synchronization trajectory leader nonlinear dynamics described leader node state nonlinear function leader defining local neighborhood synchronization error function node aij pinning gains aij aij agent directed state agent least one agent directed toward leader state global error dynamics described diag notice proof equation found remark communication graph considered strongly connected thus least one irreducible diagonally dominant hence nonsingular remark see agent state leader state rnn equation kronecker product derivative error dynamics strong connected graph minimum singular value case systems equation reflects coupling created synchronization different states agent thus interactions longer guarantee error dynamics agent confined within desired performance functions based knowledge sign prescribed performance function prescribed performance function ppf defined time function tool tracking error starts within predefined large set decays systematically predefined small set ppf developed provide smooth tracking response sufficient range control signal guarantees transient tracking performance prescribed characteristics performance function associated error component defined smooth function positive decreasing function lim ppf written exp appropriately defined positive constants order overcome difficulty caused synchronization algorithm achieve desired prescribed performance following time varying constraints proposed remark dynamic constraints represent modification ones papers constraints conditioned follows due interaction agents dynamics constraints lead instability figure illustrates tracking error case systems may exceed lower upper bounds plot green color upon crossing reference system becomes unstable cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems original formulation however switching based provides necessary control keep system stable figure shows full idea tracking error prescribed performance transits large smaller set accordance equations figure shows error trapped systematically within large set predefined small set fig graphical representation tracking error prescribed performance prescribed performance prescribed performance order transform error nonlinear system constraints unconstrained consider following relation transformed error given equivalently transformed error one notice previous set equations exchange values depending sign one note highest value involves subtracting absolute value lowest involves addition absolute value accordingly recalling equations rewritten thus transformed error expressed compact form follows sign attenuate effect chattering following form transformed error proposed erf smooth function inverse addition selected following properties satisfied erf property smooth strictly increasing remark primary role make erf close possible sign ideally selected big possible instance therefore however error function derivative smooth bounded one selects high gain risk chattering known positive constants selected according following relations exp exp exp exp exp exp exp exp consider general form smooth function exp exp exp exp parameter design simplification let define algebraic manipulations derivative transformed error approximated remark mentioned earlier selection high gain make absolute value error function converge small ratio analysis use show control generate uub error dynamic converge ball around zero radius made small desired depending selection thus error may converge zero cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems let global synchronization transformed error obtained control level node form value clarified later paper represents necessary control actions tackle uncertainties approximation errors see diag diag diag noted define dynamic boundaries initial large final small set significant impact control effort small error tracking bounded higher values require time systematic convergence large small set whereas impact noticed speed convergence large small sets implies values direct impact range control signal proceeding following definitions needed see definition global neighborhood error uniformly ultimately bounded uub exists compact set exists bound time independent definition control node trajectory given cooperative uub respect solutions node dynamics exists compact set exist bound time independent neuro adaptive distributed control prescribed performance section presents neural approximation unknown nonlinearities model uncertainties group heterogeneous agents distributed control design formulated local node satisfy prescribed characteristics neural approximations unknown nonlinearities local agents approximated rvi sufficient number neurons node rvi approximated error according neural network approximated variety sets including radial basis functions centers widths sigmoid functions etc tracking local performance node attained compensation unknown nonlinearities using available information neighbor state agents therefore nonlinearities local nodes approximated rvi approximation one mention crucial well known property structure adopted details reader invited see property continuous function every exist bounded integer optimal synaptic weight vector rvi compact set global synchronization graph described diag global estimate diag error estimating nonlinearities described parameter estimation error owing property exist unknown constant adaptive control design distributed agents following standard assumptions required assumption leader states bounded leader unknown variable dynamics bounded unknown disturbances bounded kwk variable bounded lemma let irreducible matrix nonsingular define diag diag matrix defined also positive definite cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems gist idea diagonally strictly dominant since symmetric positive definite based lemma following preposition holds proposition let positive definite diagonal matrix defined lemma matrix defined positive definite proof since nonsingular diagonal let diag means strict diagonal dominance symmetric strictly diagonally dominant therefore positive definite control signal local nodes given control gain let control signal nodes let local node tuning laws given theorem distributed adaptive control protocol synchronization prescribed performance consider strong connected graph networked system assumption distributed adaptive control protocol neighborhood synchronization errors given let adaptive estimate defined local node tuning laws follow ivi scalar gains defined satisfy transformed error written consider following lyapunov candidate function positive diagonal matrix defined lemma block diagonal matrix defined diag derivative let one write hand implies one note positive definite diagonal matrix lim rvi ivi scalar gains proof view error function defined max equation becomes therefore control node trajectory cooperative uub nodes synchronize close cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems using weight tuning law since diagonal one also kzk one define positive definite khk kzk according equivalent hence one find following relation selecting leads positive condition therefore uub consequently sufficient conditions kwm kwm leads using one kzk krk therefore contained compact set results hold assumption per property therefore need establish proposed control law initial conditions force get compact set analysis follows slight modification let suppose shows owing property minimum maximum values respectively define appropriate variables written kzk consequently equation written let kwm hence expressed follows state contained times compact set completes proof remarked associated nonlinearity compensation adaptive estimate controls speed convergence desired tracking output selected satisfy finally algorithm nonlinear single node dynamics equation summarized briefly define control design parameters evaluate local error synchronization equation evaluate ppf equation evaluate equation evaluate transformed error equation evaluate control signal equation evaluate estimate equation step remark problem extended easily estimated weight written cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems simulation results example consider problem strongly connected network composed nodes one leader connected node fig time sec agents time sec fig output performance control signal example using transformed error equation leader fig strongly connected graph one leader five agents nonlinear dynamics graph cos cos error control signal response illustrated fig respectively also fig compares proposed distributed control ppf neuroadaptive distributed control without ppf design parameters initial conditions leader response proposed algorithm illustrates high tracking performance low control effort whereas control shows instability cos cos cos represent random constants time instant leader dynamics desired consensus value following parameters used simulation ivi fig shows output performance control signal transformed error respectively proposed control algorithm using transformed error instead figures denotes output response leader node denotes control signal leader node output performance control signal associated agent respectively fig illustrates quality tracking falls constraints ppf also noticed error obeys prescribed error boundaries started within large set ended within small predefined set however control input transformed error completely oscillatory proposed distributed control ppf improved handle problem oscillation control input transformed error fig using transformed error fig present output performance control signal transformed error respectively using proposed control algorithm smooth error function erf transformed error advantage introducing smooth function reflected good tracking performance oscillation behavior transformed fig output performance control signal example control ppf using transformed error equation compared control fig error transformed error control ppf example using transformed error equation example consider problem fig inputs outputs nonlinear systems nonlinear dynamics defined cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems cos sin sin cos time sec time sec fig control ppf output performance example using transformed error equation sin sin cos cos sin time sec time sec fig error transformed error output example using transformed error equation time sec leader dynamics initial conditions equal desired consensus number neurons control parameters problem defined initial conditions time sec fig error transformed error output example using transformed error equation time sec robustness proposed controller time varying uncertain parameters time varying external disturbances high nonlinearities tested example fig shows output performance control input proposed controller mimo case tracking errors associated transforms depicted fig control signal smooth dynamic performance satisfy prescribed dynamic constraints time sec fig error transformed error output example using transformed error equation cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems conclusion distributed cooperative control proposed based prescribed performance multiagent systems neural network used estimate system nonlinearities controller developed strongly connected network control signal chosen properly ensure graph stability well track leader trajectory error subject dynamic prescribed constraints controller successfully enables agents synchronize desired trajectory short time small residual error simulation examples presented considering highly nonlinear heterogeneous systems time varying parameters uncertainties disturbances first order nonlinear systems considered higher order nonlinear systems considered future acknowledgment author would like acknowledge support provided king abdulaziz city science technology kacst science technology unit king fahd university petroleum minerals kfupm funding work project part national science technology innovation plan references fax murray information flow cooperative control vehicle formations ieee transactions automatic control vol chopra spong control multiagent systems advances robot control springer lewis zhang das cooperative control systems optimal adaptive design approaches springer science business media iii fax murray consensus cooperation networked systems proceedings ieee vol ren beard distributed consensus cooperative control springer zhang lewis adaptive cooperative tracking control nonlinear systems unknown dynamics automatica vol das lewis distributed adaptive control synchronization unknown nonlinear networked systems automatica vol cao ren distributed coordinated tracking reduced interaction via variable structure approach ieee transactions automatic control vol hou cheng tan decentralized robust adaptive control multiagent system consensus problem using neural networks ieee transactions systems man cybernetics part cybernetics vol cheng hou tan lin zhang adaptive control multiagent systems uncertainties ieee transactions neural networks vol theodoridis boutalis christodoulou direct adaptive trajectory tracking uncertain nonlinear systems international journal adaptive control signal processing vol qureshi lewis cooperative tracking control unknown affine nonlinear systems automatica vol lewis jagannathan yesildirak neural network control robot manipulators systems crc press liu guan pulsemodulated intermittent control consensus multiagent systems ieee transactions systems man cybernetics systems bechlioulis rovithakis robust adaptive control feedback linearizable mimo nonlinear systems prescribed performance ieee transactions automatic control vol hashim lewis adaptive synchronisation unknown nonlinear networked systems prescribed performance vol taylor francis bechlioulis rovithakis adaptive control guaranteed transient steady state tracking error bounds strict feedback systems automatica vol tong sui fuzzy adaptive output feedback control mimo nonlinear systems partial tracking errors constrained ieee transactions fuzzy systems vol tong prescribed performance adaptive fuzzy dynamic surface control nonlinear largescale systems time delays information sciences vol tong liu feng adaptive control design prescribed performance switched nonlinear systems automatica vol karayiannidis papageorgiou doulgeri modelfree controller guaranteed prescribed performance tracking robot joint positions velocities ieee robotics automation letters vol adaptive prescribed performance control nonlinear systems unknown dead zone international journal adaptive control signal processing vol yang wang hua adaptive neural network based prescribed performance control teleoperation system input saturation journal franklin institute vol wang hovakimyan cao verifiable adaptive flight control unmanned combat aerial vehicle aerial refueling journal guidance control dynamics vol kostarigka rovithakis adaptive dynamic output feedback neural network control uncertain mimo nonlinear systems prescribed performance ieee transactions neural networks learning systems vol mohamed improved robust adaptive control highorder nonlinear systems guaranteed performance king fahd university petroleum minerals wang chen pinning complex dynamical network equilibrium circuits systems regular papers ieee transactions vol iii khoo xie man robust consensus tracking algorithm multirobot systems transactions mechatronics vol iii cooperative control dynamical systems applications autonomous vehicles springer science business media wang yang dynamic learning adaptive neural control robot manipulators prescribed performance ieee transactions systems man cybernetics systems cite article sami hashim hashim frank lewis distributed control prescribed performance synchronization unknown nonlinear networked ieee transactions systems man cybernetics systems hornik stinchcombe white multilayer feedforward networks universal approximators neural networks vol poggio girosi regularization algorithms learning equivalent multilayer networks science vol cotter theorem application neural ieee transactions neural publication ieee neural networks council vol ioannou robust adaptive control class mimo nonlinear systems guaranteed error bounds ieee transactions automatic control vol
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arxiv nov semantics programming languages jan heering paul klint cwi cwi university amsterdam box amsterdam netherlands box amsterdam netherlands abstract paying attention tool generation programming language semantics significantly increase impact ultimately may lead language design assistants incorporating substantial amounts semantic knowledge role programming language semantics programming language semantics lost touch large groups potential users among reasons unfortunate state affairs one stands semantic results rarely incorporated practical systems would help language designers implement test language development assist programmers answering questions meaning language feature properly documented language reference manual nevertheless systems potentially effective bringing formalisms techniques places needed dissemination publications courses even exemplary programming languages current situation semantics languages tools drifting steadily apart shown figure approach semantics aims making semantics definitions useful productive generating many tools possible expect reverse current trend shown figure goal produce semantically wellfounded languages tools ultimately envision emergence language design assistants incorporating substantial amounts semantic knowledge table lists semantics definition methods aware examples use found petri nets process algebras methods specifically address semantics programming languages included dating back sixties attribute grammars denotational semantics among oldest methods abstract state machines formerly called evolving algebras coalgebra semantics program algebra latest additions field ironically attribute grammars popular tool builders semanticists consider particularly interesting definition method since discuss various methods general terms without going technical details reader need familiar case differences often hard decipher field highly fragmented appropriate dictionaries lacking affect main argument table lists representative language development system semantics definition methods table last entry software refinery origins software research supported part telematica instituut languages project semantics languages tools figure semantics languages tools drifting steadily apart semantics languages tools figure approach semantics languages tools kept together tool generation ultimately language design assistants ldas semantics axiomatic attribute grammars denotational algebraic structural operational natural action abstract state machines coalgebraic program algebra definition terms postconditions attribute propagation rules rules inference rules action expressions transition rules behavioral specification rules equations table current approaches programming language semantics semantics attribute grammars denotational algebraic structural natural action abstract state machines operational system synthesizer generator psg centaur developed cornell university technical university darmstadt cwi university amsterdam asd software refinery cwi university aarhus university aquila reasoning systems palo alto inria table representative language development systems environments research kestrel institute fit current semantics paradigms pioneering semanol system best knowledge longer use included systems listed widely different capabilities widely different stages development discussing characteristics applications section first explain general ideas underlying approach programming language semantics shaped experiences table past ten years finally discuss language design assistants section approach semantics approach semantics aims making semantics definitions useful productive generating many tools possible affects many aspects way programming language semantics practiced upsets dogmas table lists tools might generated principle language definition augmented suitable information tool generated may require language extensions core semantics definition formalism practice always necessary since semantics definitions tend contain good deal implicit information may extracted used tool generation prettyprinter editor typechecker abstract interpreter dataflow analyzer call graph extractor partial evaluator optimizer program slicer origin tracker debugger code generator compiler profiler test case generator test coverage analyzer regression test tool complexity analyzer metrics documentation generator cluster analysis tool systematic program modification tool table tools might derived language definition first entry table scanner parser generation standard technology lex yacc examples generators purpose input formalisms close regular expressions bnf facto standard formalisms regular grammars respectively unfortunately tools table standard formalisms key features approach language definitions primarily tool generator input provide kind theoretical explanation constructs language question become part language reference manual interpreter act among things oracle programmers needing help among first tools generated writing large language definitions loses esoteric character becomes similar kind programming semantics formalisms tend best small examples lose much power language definitions written grow approach semantics formalisms modular separate generation analogue separate compilation supported libraries language constructs become important approach may require addition features core formalism leads rather pure style semantics description scope approach includes instance system generated tools synthesizer generator lalr prettyprinter editor incremental typechecker incremental translator psg asd editor incremental typechecker even incomplete program fragments interpreter generalized prettyprinter syntaxdirected editor typechecker interpreter origin tracker translator renovation tools lalr prettyprinter editor typechecker interpreter origin tracker translator editor checker interpreter typechecker interpreter debugger software refinery lalr prettyprinter editor parse tree repository including dataflow relations tools program slicer metaenvironment centaur semantic engine incremental attribute evaluator functional language interpreter conditional rewrite rule engine inference rule engine conditional rewrite rule engine transition rule engine tree manipulation engine table tool generation capabilities representative language development systems little languages many tools table useful dsls programming languages software maintenance renovation tools included end table compiler toolkits cosy cocktail ocs suif pim existing language development systems table summarizes tool generation capabilities representative language development systems listed table generate lexical scanners parsers prettyprinters many produce editors typecheckers interpreters produce various kinds software renovation tools end support one specification formalisms differ generality application domain instance synthesizer generator supports attribute grammars incremental attribute evaluation particularly suitable typechecking static analysis translation less suitable dynamic semantics supports conditional rewrite rules rather attribute grammars used defining dynamic semantics well software refinery comes functional language wide range computations programs expressed systems provide specialized specification formalisms psg instance uses context relations describe incremental typechecking even incomplete program fragments denotational definitions dynamic semantics supports formalism optimized definition programming language semantics tool generation generate typechecker interpreter debugger table far complete language development systems sis psp gag sps mess actress pregmatic ldl eli many tools listed table generated current system ample opportunities tool generation still exist areas like optimization dynamic program analysis testing maintenance toward language design assistants logical next step beyond tool generation would lead situation similar computer algebra large parts mathematics incorporated computer algebra systems conversely computer algebra become fruitful mathematical activity yielding new results general mathematical interest case semantics see opportunities language design assistants incorporating substantial amount formal informal semantic knowledge latter found instance language design rationales discussion documents produced standardization bodies development assistants push semantics even toward practical application also give rise new theoretical questions language design assistants mind would support human language designer providing design choices performing consistency checks design process operational knowledge typical issues like typing rules scope rules execution models incorporated major research questions arise regarding acquisition representation organization abstraction level required knowledge instance organized according currently known paradigms functional logic programming higher level abstraction found new paradigms derived constraints composition certain features expressed checked another key question construct collection language feature components sufficiently general reusable across wide range languages similar considerations apply tool development incorporating knowledge tool generation language design assistant envision tool generation assistant helps constructing tools advanced way tool generation mind previous sections make perspective somewhat tangible consider relatively simple case conditional construct modelled language feature component table gives impression wide range issues addressed generic conditional construct specialized concrete conditional expression specific language research question design abstract framework similar questions expressed answered another major question organize specialization process language feature component concrete language construct main alternatives parameterization transformation using parameterization specialization component question amounts instantiating parameters since parameters identified beforehand instantiation usually rather simple mechanism parameterized component limited using transformations hand language feature component designed without explicit parameters specialization achieved applying appropriate transformation rules obtain desired specific case clearly approach flexible since part language feature component modified transformation rules thus effectively act parameter relation approach transformation parameterized modules largely unexplored type expression controlling selection one two branches controlling expression evaluated short circuit full evaluation controlling expression evaluated concurrently program parts speculative execution conditional special case controlling expression controlling expression cause exceptions jumps outside branches allowed selected branch evaluated concurrently program parts evaluation selected branch cause evaluation selected branch cause exceptions evaluation conditional construct yield value table possible parameters generic conditional construct although aware research language design assistants broad perspective sketched work pointing general direction language designer workbench sketched future work goals action semantics also emphasizes libraries reusable language constructs plans longer pursued language development laboratory included library reusable language constructs knowledge base containing knowledge languages tool language design design implementation selection languages described case study interactive composition predefined language constructs acknowledgements would like thank jan bergstra mark van den brand arie van deursen ralf jan rutten useful comments earlier versions references ace cosy compilation system anderson belz blum semanol metalanguage programming semantics programming languages acta informatica anlauff kutter pierantonio formal aspects development environments montages sellink 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hierarchical deep learning architecture objects classification atul laxman krishna prasad amish kumar sehaj singh mynepalli siva samsung institute india bangalore bagmane constellation business park doddanekundi circle bangalore india abstract evolution visual object recognition architectures based convolutional neural networks convolutional deep belief networks paradigms revolutionized artificial vision science architectures extract learn real world hierarchical visual features utilizing supervised unsupervised learning approaches respectively approaches yet scale realistically provide recognition large number objects high propose two level hierarchical deep learning architecture inspired divide conquer principle decomposes large scale recognition architecture root leaf level model architectures root leaf level models trained exclusively provide superior results possible deep learning architecture prevalent today proposed architecture classifies objects two steps first step root level model classifies object high level category second step leaf level recognition model recognized high level category selected among leaf models leaf level model presented input object image classifies specific category also propose blend leaf level models trained either supervised unsupervised learning approaches unsupervised learning suitable whenever labelled data scarce specific leaf level models currently training leaf level models progress trained total leaf level models trained leaf models best case error rate validation data set particular leaf models also demonstrate validation error leaf level models saturates towards mentioned accuracy number epochs increased sixty error rate entire architecture needs computed conjunction error rates root leaf models realization two level visual recognition architecture greatly enhance accuracy large scale object recognition scenarios demanded use cases diverse drone vision augmented reality retail image search retrieval robotic navigation targeted advertisements etc keywords convolutional neural network cnn convolutional deep belief network cdbn supervised unsupervised training introduction deep learning based vision architectures learn extract represent visual features model architectures composed layers transformations stacked top learn high level abstractions low level features extracted images utilizing supervised unsupervised learning algorithms recent advances training cnns gradient descent based backpropagation algorithm shown accurate results due inclusion rectified linear units nonlinear transformation also extension unsupervised learning algorithms train deep belief networks towards training convolutional networks exhibited promise scale realistic image sizes supervised unsupervised learning approaches matured provided architectures successfully classify objects categories respectively yet approaches scaled realistically classify objects categories need large scale object recognition ever relevant today explosion number individual objects supposed comprehended artificial vision based solutions requirement pronounced use case scenarios drone vision augmented reality retail image search retrieval industrial robotic navigation targeted advertisements etc large scale object recognition enable recognition engines cater wider spectrum object categories also mission critical use cases demand higher level accuracy simultaneously large scale objects recognized paper propose two level hierarchical deep learning architecture achieves compelling results classify objects categories best knowledge proposed method first attempt classify objects utilizing two level hierarchical deep learning architecture also blend supervised unsupervised learning based leaf level models proposed overcome labelled data scarcity problem proposed architecture provides dual benefit form providing solution large scale object recognition time achieving challenge greater accuracy possible deep learning architecture related works come across work uses hierarchical deep learning architecture classify objects images object recognition large scale using shallow architectures utilizing svms discussed effort presents study large scale categorization image classes using spatial pyramids spm bag visual words bow feature extraction support vector machines svm classification work creates ten different datasets derived imagenet categories based datasets outlines influence classification accuracy due different factors like number labels dataset density dataset hierarchy labels dataset methods proposed provide extended information classifier relationship different labels defining hierarchical cost cost calculated height lowest common ancestor wordnet classifiers trained loss function using hierarchical cost learn differentiate predict similar categories compared trained loss error rate classification entire categories conclusively stated work problem statement supervised learning based deep visual recognition cnn architectures composed multiple convolutional stages stacked top learn hierarchical visual features captured figure regularization approaches stochastic pooling dropout data augmentation utilized enhance recognition accuracy recently faster convergence architectures attributed inclusion rectified linear units relu nonlinearity layer weights state art top error rate reported classification categories utilizes mentioned generic architectural elements layers weights unsupervised learning based architecture model convolutional dbn learns visual feature hierarchy greedily training layer layer architectures reported accuracy classifying objects architectures yet scaled classification objects conjecture scaling single architecture realistic computations get intractable utilize deeper architectures figure learning hierarchy visual features cnn architecture proposed method employ divide conquer principle decompose classification root leaf level distinct challenges proposed architecture classifies objects two steps captured root level model architecture first step root first level architectural hierarchy recognizes high level categories deep vision architectural model weight layers trained using stochastic gradient descent architectural elements captured table leaf level model architecture second step leaf level recognition model recognized high level category selected among leaf models leaf level model presented input object image classifies specific category leaf level architecture architectural hierarchy recognizes specific objects finer categories model trained using stochastic gradient descent architectural elements captured table cdbn based leaf level models trained unsupervised learning approach case scarce labelled images deliver blend leaf models trained supervised unsupervised approaches root level model leaf level models need trained use dataset compiled synsets release imagenet leaf node least labelled images amount images total figure two levels hierarchical deep learning archtecture supervised training vision low level features pixels etc assembled form high level abstractions edges motifs higher level abstractions turn assembled form higher level abstractions object parts objects substantial number recognition models hierarchical architecture trained utilizing supervised training algorithms utilized method referred propagation algorithm algorithms require significantly high number labelled training images per object category data set cnn based architecture cnn biologically inspired architecture multiple trainable convolutional stages stacked top cnn layer learns feature extractors visual feature hierarchy attempts mimic human visual system feature hierarchy manifested different areas human brain eventually fully connected layers act feature classifier learn classify extracted features cnn layers different categories objects fully connected layers likened area brain classifies hierarchical features generated area root level leaf level cnn models architecture trained supervised gradient descent based backpropagation method learning method cross entropy objective function minimized error correction learning mechanism computes gradients weight updates hidden layers recursively computing local error gradient terms error gradients next connected layer neurons correcting synaptic weights free parameters network eventually actual response network moved closer desired response statistical sense table cnn architecture layers architectural elements root level visual recognition architecture layer type input size kernels convolutional max pool convolutional convolutional max pool convolutional convolutional max pool convolutional convolutional convolutional max pool convolutional convolutional convolutional max pool convolutional architectural elements architectural elements proposed architecture enhanced discriminative function chosen deeper architectures smaller kernels root leaf models make objective function discriminative interpreted making training procedure difficult making choose feature extractors higher dimensional feature space relu utilized relu nonlinearities sigmoidal nonlinearities layer reduces training time converging upon weights faster pooling output combination fed pooling layer alternative convolutional layers output pooling layer invariant small changes location features object pooling method used either maxpooling stochastic pooling max pooling method averages output neighborhood neurons poling neighborhoods overlapping majority leaf models used max pooling approach overlapping neighborhoods alternatively also used stochastic pooling method training models stochastic pooling output activations pooling region randomly picked activations within pooling region following multinomial distribution distribution computed neuron activation within given region approach hyper parameter free cnn architecture stochastic pooling technique captured table dropout method output neuron fully connected layer set zero probability ensures network samples different architecture new training example presented besides method enforces neurons learn robust features rely existence neighboring neurons table cnn architecture layers architectural elements leaf level visual recognition architecture max pooling strategy layer type input size kernels convolutional max pool convolutional convolutional max pool convolutional convolutional max pool convolutional convolutional max pool convolutional table cnn architecture layers architectural elements leaf level visual recognition architecture stochastic pooling strategy layer type input size kernels convolutional stochastic pool convolutional convolutional stochastic pool convolutional convolutional stochastic pool convolutional convolutional stochastic pool convolutional training process modified libccv open source cnn implementation realize proposed architecture trained nvidia titan gpus root leaf level models trained using stochastic gradient descent leaf models trained batches models per gpu two gpu systems simultaneously first leaf models initialized trained scratch epochs learning rate momentum rest leaf models initialized trained leaf models trained learning rate root model trained epochs learning rate initialized similar model trained imagenet dataset takes days batch leaf models train epochs currently root model leaf models trained weeks full realization architecture progress estimated conclude second week september unsupervised training statistical mechanics inspired concept unsupervised training fundamentally specifically statistical mechanics forms study macroscopic equilibrium properties large system elements starting motion atoms electrons enormous degree freedom necessitated statistical mechanics foundation makes use probabilistic methods suitable candidate modelling features compose training data sets cdbn based architecture networks trained statistical mechanics fundamentals model underlying training dataset utilizing boltzmann distribution obviate painfully slow training time required train boltzmann machines multiple variants proposed restricted boltzmann machine rbm one provided best possible modelling capabilities minimal time resulting stacks rbm layers greedily trained layer layer resulting deep belief networks dbn successfully provides solution image speech recognition document retrieval problem domains dbn described multilayer generative models learn hierarchy feature detectors lower layer learns lower level features feeds higher level help learn complex features resulting network maximizes probability training data generated network dbn limitations scaling realistic image sizes first difficulty computationally tractable increasing image sizes second difficulty faced lack translational invariance modelling images scale dbn modelling realistic size images powerful concept convolutional dbn cdbn introduced cdbn learns feature detectors translation invariant feature detectors detect features located location image perform block gibbs sampling using conditional distribution suggested learn convolutional weights connecting visible hidden layers activations neurons visible hidden layers respectively also hidden unit biases visible unit biases forms weight matrix connecting visible hidden layer gibbs sampling conducted utilizing weights learnt give layers convolutional rbms crbm crbms stacked top form cdbn probabilistic max pooling layers convolutional layers mlp introduced top complete architecture concept captured figure train first two layers leaf architecture unsupervised learning later abandon unsupervised learning use learnt weights first two layers initialize weights cnn architecture cnn architecture weights using backpropagation method architecture used training unsupervised learning mechanism captured table also hierarchical deep learning architecture constructed entirely cdbn depicted figure figure convolutional dbn constructed stacking crbms learnt one one training process used contrastive divergence mechanism train first two layers architecture specified unsupervised learning updates hidden units positive phase step done sampling rather using real valued probabilities also used size training monitored accuracy training utilizing reconstruction error refers squared error original data reconstructed data guarantee accurate training course training generally decrease also large amount increase suggests training going wrong printing learned weights learned weighs needs eventually visualized oriented localized edge filters printing weights training helps identify whether weights approaching shape ratio variance reconstructed data variance input image exceeds decrease learning rate factor reset values weights hidden bias updates ensure weights explode initial momentum chosen increased finally initial learning rate chosen figure proposed level hierarchical deep learning architecture constructed entirely utilizing cdbns objects hierarchy creation hierarchy design classification objects categories requires decision making following parameters number leaf models trained number output nodes leaf model decide parameters first build hierarchy tree synsets classes dataset described section using set described section try split organize classes leaf models holding maximum classes building hierarchical tree using wordnet relationship synsets dataset organized hierarchical tree wordnet isa relationship file lists relationships synsets imagenet example line relationship file refers parent synset cow child synset heifer however isa file relate single child multiple parents heifer also child another category young mammal depth synset imagenet hierarchy relationship semantic label focused building deepest branch synset utilized simplified method exploits relationship synset depth deeper category nxxx larger number xxx hence used parent category heifer cow instead young mammal algorithm htree depicted figure used generate hierarchy tree paper results ninth iteration used base tree sample illustration captured figure building hierarchical tree hierarchy tree evident dataset skewed towards categories like flora animal fungus natural objects instruments etc levels closer tree root synsets fall branches figure hierarchy tree generation htree algorithm figure hierarchy tree generation htree algorithm figure hierarchy tree generated htree algorithm iterations categories part model taking account number models trained time resources required model decided finalize solution parameters ideal hierarchy leaf models capable classifying categories synsets root leaf level models decided hierarchy tree flat possible total number synsets leaf model less leaf level models remaining subcategories split merged another leaf final solution parameters final solution parameters follows leaf models leaf model classifying synsets results formulate root leaf models hierarchy root level model leaf level models need trained leaf level model recognizes categories range numbers use dataset compiled synsets release imagenet leaf node least labelled images amount images total top error rates leaf level models computed graph fig plots top errors leaf models training epochs observe leaf models trained higher number training epochs top error decreases top error rate complete objects classification computed upon training models required hierarchical deep learning architecture figure graph captures decrease error rate increase increase number epochs training supervised training method training classification objects proposed hierarchical architecture progress estimated completed mid september architecture root model leaf models based cnn architecture trained supervised gradient descent utilizing unsupervised learning trained leaf model consists artifacts categories model cdbn based architecture described section trained first layer cdbn architecture contrastive divergence algorithm later first layer weights utilized initialize weights model supervised setting utilizing supervised learning feature extractors kernels learnt supervised unsupervised learning captured fig intend compute top error rates categories using algorithm captured fig figures depict classification results using hierarchical deep learning architecture first image original image used testing remaining images represent categories predicted hierarchical model descending order confidence conclusions error rate entire architecture required computed conjunction error rates root leaf models realizations two level visual recognition architecture greatly simply object recognition scenario large scale object recognition problem time proposed two level hierarchical deep learning architecture help enhance accuracy complex object recognition scenarios significantly otherwise would possible architecture figure left image depicts first layer numbers filter weights learned utilizing right image depicts first layer weights back propagation trade proposed architecture size hierarchical recognition model total size recognition models including root leaf models amounts approximately size might put constraints towards executing entire architecture devices size constraint executed high end device cloud based recognition ram size higher besides always split hierarchical model device cloud paves way object recognition utilizing novel collaboration architectures proposed architecture soon available classifying large scale objects breakthrough help enable applications diverse drone vision industrial robotic vision targeted advertisements augmented reality retail robotic navigation video surveillance search information retrieval multimedia content etc hierarchical recognition models deployed commercialized various devices like smart phones home gateways head set various use cases figure algorithm compute top error rates categories evaluated hierarchical deep learning algorithm figure test image belongs wilson warbler predicted categories order confidence yellow warbler wilsons warbler yellow hammer wren warbler audubon warbler figure test image belongs fruit kumquat predicted categories order confidence apricot loquat fuyu persimmon golder delicious kumquat figure test image belongs racing boat predicted categories racing boat racing shell outrigger canoe gig rowing boat figure test image belongs category oak predicted categories live oak shade tree camphor spanish oak acknowledgements take opportunity express gratitude deep regards mentor shankar venkatesan guidance constant encouragement without support would possible materialize paper references yann lecun koray kavukvuoglu farabet convolutional networks applications vision proc international symposium circuits systems ieee krizhevsky ilya sutskever hinton imagenet classification deep convolutional neural networks nips simonyan zisserman deep convolutional networks image recognition arxiv technical report lee grosse ranganath convolutional deep belief networks scalable unsupervised learning hierarchical representations proceedings international conference machine learning montreal canada jia deng alexander berg kai classifying image categories tell computer sergey ioffe christian szegedy batch normalization accelerating deep network training reducing internal covariate shift deng dong socher imagenet hierarchical image database george dahl dong deng alex acero deep neural networks speech recognition ieee transactions audio speech language processing haykin neural networks comprehensive foundation new jersey prentice hall orban higher order visual processing macaque extrastriate cortex physiological reviews vol doi modern computer vision library online http matthew zeiler rob fergus stochastic pooling regularization deep convolutional neural networks iclr dance willamowski fan bray csurka visual categorization bags keypoints eccv international workshop statistical learning computer vision lazebnik schmid ponce beyond bags features spatial pyramid matching recognizing natural scene categories ieee computer society conf computer vision pattern recognition authors atul laxman katole atul laxman katole completed jan signal processing indian institute science bangalore currently working samsung institute technical areas interest include artificial intelligence deep learning object recognition speech recognition application development algorithms seeded established teams domain deep learning applications image speech recognition krishna prasad yellapragada krishna prasad yellapragada currently working technical lead samsung institute interests include deep learning machine learning content centric networks amish kumar bedi amish kumar bedi completed computer science iit roorkee batch working samsung institute indiabangalore since july technical areas interest include deep learning artificial intelligence sehaj singh kalra sehaj sing kalra completed computer science iit delhi batch working samsung institute india bangalore since june interest lies machine learning applications various domains specifically speech image also find artificial intelligence nlp data mining interesting mynepalli siva chaitanya mynepalli siva chaitanya completed electrical engineering iit bombay batch working samsung institute india bangalore since june area interest includes neural networks artificial intelligence
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pedotransfer functions reliability estimating saturated hydraulic conductivity behzad vahid yakov bureau economic geology jackson school geosciences university texas austin austin physiosigns sunnyvale environmental microbial safety beltsville corresponding author email address abstract saturated hydraulic conductivity ksat fundamental characteristic modeling flow contaminant transport soils sediments therefore many models developed estimate ksat easily measureable parameters textural properties bulk density etc however ksat affected textural structural characteristics also sample dimensions internal diameter height using unsoda database contrast pattern aided regression cpxr method recently developed sample pedotransfer functions estimate ksat textural data bulk density sample dimensions main objectives study evaluating proposed pedotransfer functions using larger database comparing seven models purpose selected nineteen thousands soil samples around united states results showed sample pedotransfer functions estimated ksat accurately seven models frequently used literature keywords contrast pattern aided regression pedotransfer function sample dimensions saturated hydraulic conductivity introduction saturated hydraulic conductivity ksat essential modeling surface subsurface flow well solute transport soils sediments estimation intensively investigated past several decades soil physicists hydrologists many empirical hazen puckett nemes parasuraman kozeny carman ahuja rawls arya timlin katz thompson skaggs porter hunt ghanbarian models well numerical techniques see zhang elliot mostaghimi ghanbarian daigle dal ferro morari developed estimate ksat available properties among pedotransfer functions regressionbased models making use available information provide estimates related factors needed warrick attracted great deal attention mainly due relatively high efficiency simplicity pedotransfer functions also limitations example schaap leij showed performance pedotransfer functions may strongly depend data used train develop test validate nonetheless pedotransfer functions actively developed link soil hydraulic properties readily measureable characteristics soil texture bulk density organic matter content estimate saturated hydraulic conductivity soil parameters pedotransfer functions developed using various techniques multiple linear regression brakensiek cosby saxton jabro artificial neural networks schaap leij schaap parasuraman genetic programming parasuraman decision tree analysis shirley jorda group method data handling nemes support vector machines twarakavi contrast pattern aided regression ghanbarian minasny mcbratney evaluated eight proposed pedotransfer functions estimation saturated hydraulic conductivity ksat using published australian soil data sets comprised different field laboratory measurements large areas limited predictive variables found published ptfs dane puckett cosby schaap gave best estimations sandy loamy clayey soils respectively sobieraj evaluated performance nine ksat pedotransfer functions including rosetta model see table modeling storm flow generated rainforest catchment found inaccurate shallow depths pedotransfer functions except jabro commonly underestimated ksat subsequent depths pedotransfer functions except campbell shiozawa typically overestimated ksat gwenzi evaluated five ptfs including cosby jabro puckett dane puckett saxton showed evaluated ksat models underestimated order magnitude suggesting application water balance simulation models based ptfs may constitute bias model outputs recently ghanbarian proposed sample pedotransfer functions estimate saturated hydraulic conductivity ksat using contrast pattern aided regression cpxr method soil samples unsoda database showed including sample dimensions sample internal diameter height length substantially improved ksat estimations although numerous studies assessed performance different pedotransfer functions estimation water content soil water retention curve evaluations ksat pedotransfer functions estimations limited literature particularly restricted assessments small scales therefore main objective study use large database including soil samples around united states evaluating sample pedotransfer functions developed ghanbarian using contrast pattern aided regression cpxr method estimation saturated hydraulic conductivity ksat clay silt sand contents geometric mean diameter standard deviation diameters soil particles bulk density sample dimensions internal diameter height comparing performance proposed pedotransfer functions performance seven pedotransfer functions frequently applied literature materials methods first describe datasets used study present proposed pedotransfer functions available literature estimating saturated hydraulic conductivity clay silt sand content well bulk density datasets entire database used study includes datasets soil samples selected usksat database pachepsky park different states nation number samples ksat measurement method average standard deviation different parameters dataset summarized table average ksat value varying jabro romkens near three orders magnitude variation indicates different types soils various structures order magnitude greater average ksat range schaap investigated soil samples sources united states results discussion section however show measured ksat value actually spans even much orders magnitude representing wide range soils figure showing percentage samples usda soil texture class distribution samples ternary diagram also implies various types soils different textures used study saturated hydraulic conductivity pedotransfer functions tremendous amount models literature estimating saturated hydraulic conductivity properties however focus study ksat estimation frequently available characteristics soil textural data sand silt clay content bulk density soil mapping purposes large scales table list models commonly used literature including brakensiek campbell shiozawa cosby jabro puckett dane puckett saxton campbell shiozawa puckett used database consisting samples six ultisols taken seven locations lower coastal plain alabama cosby model proposed based samples holtan rawls jabro presented model using soil samples collected literature southern cooperation series bulletins dane puckett developed model using samples ultisols lower coastal plain alabama saxton model developed using database including selected data points uniformly spaced hydraulic conductivity curves textures classes reported rawls models except brakensiek estimate ksat sand silt clay content bulk density brakensiek model however requires porosity value well estimated particle density assumed approximately equal samples point eliminated jabro dataset see table model evaluation process since assessment jabro model using dataset used develop would biased supported addition jabro model estimates ksat logarithms clay silt content see table consequence jabro model returns unrealistic ksat estimate samples either zero clay silt content removing jabro dataset well samples either zero clay silt content resulted evaluation jabro model sample model ghanbarian presented table consists patterns corresponds specific model patterns given table match support training data ghanbarian noted even patterns supports less still match samples training data moreover patterns relatively small supports often large arr average residual reduction values implying matching samples pattern represent important data group baseline model makes large prediction errors errors reduced significantly local model see table corresponding pattern ghanbarian estimate ksat using ghanbarian model first relevant pattern data match found using corresponding local model given table ksat estimated one expect one soil sample may match one pattern case relevant patterns identified used estimate ksat weighted average ksat values estimated different patterns local models represents ghanbarian model estimation case sample match pattern ksat estimated using baseline model given table detail cpxr approach found ghanbarian java code ksat estimation ghanbarian model models presented table given supplementary material order compare statistically accuracy discussed models estimation ksat root mean square error rmsle mean error mle parameters determined follows log log log log number values ksat cal ksat meas calculated estimated measured saturated hydraulic conductivities respectively results discussion comparison performance selected models estimation saturated hydraulic conductivity ksat sand silt clay content well bulk density shown fig seen campbell shiozawa puckett models mle tend underestimate ksat considerably orders magnitude variation dane puckett saxton ghanbarian models estimates distributed less around line mle respectively interestingly although puckett campbell shiozawa models developed using exponential function dataset samples lower coastal plain alabama similar trend ksat underestimation puckett model almost accurate campbell shiozawa model rmsle discrepancy ksat estimation implies puckett model could efficiently detect nonlinear interactions input output variables campbell shiozawa model mle values different ksat models usda soil texture classes presented table also show puckett campbell shiozawa tend underestimate ksat soil texture classes particularly intermediate fine textured soils jabro model tendency overestimate ksat observed fig although value mean error cosby model near zero mle ksat estimations biased since model overestimates lower ksat soils underestimates higher ksat values soils model measured ksat spans around orders magnitude log ksat varies near estimated log ksat value restricted much narrower range approximately obtained results however consistent results minasny mcbratney found cosby model gave best ksat estimations loamy soils intermediate ksat using soil samples australia minasny mcbratney reported rmsle campbell shiozawa cosby dane puckett brakensiek saxton models respectively meaning cosby model accurate rmsle values however comparable present fig ksat units different statistical parameter calculated minasny mcbratney root mean square natural error report root mean square error number samples minasny mcbratney whereas study resulted rmsle values much smaller reported minasny mcbratney figure also shows jabro model tends overestimate ksat substantially mle although results obtained large scale might contrast results gwenzi reported jabro model underestimated ksat order magnitude number samples study compared gwenzi much larger one thus expect obtained results general rmsle values different ksat models usda soil texture classes given table bold rmsle values table indicate accurate model estimating ksat different soil texture classes precisely ghanbarian among models evaluated study shown fig estimations ghanbarian model appropriately uniformly distributed around line indicating accurate model comparison rmsle parameter calculated model shows value rmsle ghanbarian model considerably smaller determined models ghanbarian dane puckett model estimates ksat slightly better saxton cosby models interestingly dane puckett puckett models exponential function input variable clay however dane puckett model rmsle estimated ksat accurately puckett model rmsle main reason probably former soil samples used construct relationship ksat clay content latter samples consequence larger training database reliable validation results figure also shows brakensiek model estimated ksat accurately puckett campbell shiozawa models one note calculated rmsle value jabro model directly comparable determined models since model using logarithms clay silt contents therefore number samples suitable evaluate model less used assess others results study confirm sample ghanbarian model accurate estimation soil saturated hydraulic conductivity ksat main reasons discussed first model proposed ghanbarian sample means requires input variables sample height length internal diameter estimate ksat influence sample dimension ksat well documented literature example recently pachepsky showed similarity scale sample dimension dependencies soils sediments observed characteristic size increased ksat value first increased one two orders magnitude stabilized recently ghanbarian showed influence sample dimensions internal diameter height soil water retention curve estimation saturated hydraulic conductivity comparable cpxr multiple linear regression mlr methods evidence comprehensive results see ghanbarian references therein second ghanbarian model developed using novel modern data mining technique introduced recently dong taslimitehrani taslimitehrani dong dong taslimitehrani called contrast pattern aided regression cpxr models proposed using simple algorithms multiple linear regressions mlrs ghanbarian compared pedotransfer functions ones estimation ksat demonstrated circumstances cpxr method performed batter mlr technique main advantage cpxr method compared multiple linear models cpxr capable detect complex nonlinear interactions input variables define different subgroups data highly distinct response relationships extract nonlinear patterns among input output variables one advantage cpxr technique model reported readable transparent form table unlike results artificial neural networks support vector machine application visually inspected analyzed conclusion study evaluated various models available literature estimate soil saturated hydraulic conductivity ksat using large database including soil samples round united states compared reliability brakensiek campbell shiozawa cosby jabro puckett dane puckett saxton models addition models ghanbarian model estimating ksat textural data bulk density well sample dimensions height length internal diameter also assessed emphasized assessment models estimating ksat soil properties limited literature addition existing evaluations mainly restricted fairly small databases small scales therefore obtained results reported literature might contradiction results based measurements including wide range soils around nation unique general conclusively showed proposed sample pedotransfer functions ghanbarian estimated ksat substantially accurately seven frequently used previously published models acknowledgment usksat database used study available upon request yakov pachepsky publication authorized director bureau economic geology references ahuja naney green nielsen characterize spatial variability hydraulic conductivity effects land management soil sci soc ahuja cassel bruce barnes tion spatial distribution hydraulic conductivity using effective porosity data soil sci arya leij shouse van genuchten relationship hydraulic conductivity function distribution soil science society america journal bathke cassel anistropic variation profile characteristics saturated hydraulic conductivity ultisol landscape soil 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clinical prognostic modeling results using method traumatic brain injury inbioinformatics bioengineering bibe ieee international conference ieee http timlin ahuja pachepsky williams gimenez rawls use parameters improve estimates saturated conductivity effective porosity soil science society america journal twarakavi simunek schaap development pedotransfer functions estimation soil hydraulic parameters using support vector machines soil sci soc warrick soil water dynamics oxford university press usa developing new form permeability constant homogeneous porous media means fractal geometry adv water resour zhang deeks bengough crawford young determination soil hydraulic conductivity lattice boltzmann method soil technique journal hydrology zobeck fausey effect sample crosssectional area saturated hydraulic conductivity two structured clay soils trans asae table average standard deviation parentheses salient properties soil samples various references usksat database used study sample height sample diameter ksat constant head constant head dane constant head romkens constant head bathke cassel dane samples measurement method kool falling head hubbard constant head boul southard constant head robbins hydraulic head bruce peele reference bulk density sand silt clay permeameter pressure head zobeck miscellaneous jabro constant head bruce constant head habecker permeameter radcliffe constant head coelho falling head southard boul constant head rawls constant head carlisle constant head carlisle constant head baumhardt constant head romkens constant head sharratt unknown constant head price including constant head blocks constant head cores falling head permeameter table proposed pedotransfer functions frequently used literature estimate soil saturated hydraulic conductivity ksat ksat model ksat campbell shiozawa ksat cosby ksat reference brakensiek jabro ksat puckett ksat dane puckett ksat saxton ksat sand silt clay bulk density porosity table patterns criteria baseline regression model pedotransfer functions cpxr model developed ghanbarian estimate soil saturated hydraulic conductivity sand silt clay content geometric mean geometric standard deviation particles bulk density sample dimensions length internal diameter see context detail ksat estimation using baseline criteria local models log ksat baseline model criteria arr support pattern pattern local model corresponding pattern sand silt clay bulk density geometric mean diameter geometric standard deviation arr average residual reduction pedotransfer functions estimating soil saturated hydraulic conductivity shc ksat soil properties available online http table mean error mle different models used study estimate ksat usda soil texture classes usda soil texture class ksat model lsa sal sil sacl sicl sac sic mle brakensiek campbell shiozawa cosby jabro puckett dane puckett saxton ghanbarian sand lsa loamy sand sal sandy loam loam sil silt loam silt sacl sandy clay loam clay loam sicl silty clay loam sac sandy clay sic silty clay clay table root mean square error rmsle different models used study estimate ksat usda soil texture classes usda soil texture class ksat model lsa sal sil sacl sicl sac sic rmsle brakensiek campbell shiozawa cosby jabro puckett dane puckett saxton ghanbarian sand lsa loamy sand sal sandy loam loam sil silt loam silt sacl sandy clay loam clay loam sicl silty clay loam sac sandy clay sic silty clay clay bold numbers denote least rmsle values accurate model soil texture class soil texture clay silty clay sandy clay silty clay loam clay loam sandy clay loam silt silt loam loam sandy loam loamy sand sand percentage samples fig percentage samples usda soil texture class distribution samples ternary diagram log calculated ksat jabro rmsle mle puckes rmsle mle dane puckes rmsle mle ghanbarian rmsle mle log measured ksat log measured ksat log measured ksat log measured ksat log calculated ksat log calculated ksat saxton rmsle mle log measured ksat log measured ksat log measured ksat cosby rmsle mle log calculated ksat log calculated ksat campbell shiozawa rmsle mle log calculated ksat brakensiek rmsle mle log calculated ksat log calculated ksat log measured ksat fig comparison saturated hydraulic conductivity ksat estimated using eight different methods measured ones data used study note number samples elavuation jabro model samples either zero silt clay content removed entire database red dashed line represents line
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efficiently finding small separators temporal graphs philipp till hendrik rolf niedermeier feb institut softwaretechnik und theoretische informatik berlin germany zschoche abstract vertex separators vertex sets whose deletion disconnects two distinguished vertices graph play pivotal role algorithmic graph theory many realistic models real world necessary consider graphs whose edge set changes time specifically edges labeled time stamps literature graphs referred temporal graphs temporal networks networks networks etc extensive literature separators static graphs much less known temporal setting building previous work study problem finding small vertex set separator temporal graph two designated terminal vertices removal set breaks temporal paths connecting one terminal herein consider two models temporal paths paths contain arbitrarily many hops per time step nonstrict paths contain one hop per time step strict settle hardness dichotomy versus solvability problem variants regarding number time steps temporal graph moreover prove problem variants even temporal graphs whose underlying graph planar show temporal graphs whose underlying graph planar additionally number time steps constant problem variant strict paths solvable quasi linear time finally general temporal graphs introduce notion temporal core vertices whose incident edges change time prove temporal graphs temporal core variant solvable polynomial time degree polynomial independent size temporal core strict variant remains introduction complex network analysis nowadays common access process graph data interactions data set using static graphs mathematical model problem instances dynamics interactions reflected important information data might captured straightforward approach incorporate dynamics interactions model use temporal graphs temporal graph informally speaking graph edge set may change discrete time interval dynamics interactions represented model essential adapt definitions connectivity paths supported stiftung berufliche bildung sbb supported dfg project damm partially supported dfg project mate temporal graph layers figure subfigure shows temporal graph subfigure four layers gray squared vertex forms strict temporal temporal two squared vertices form temporal respect temporal nature model directly affects notion separators temporal setting vertex separators fundamental primitive static network analysis folklore computed polynomial time contrast static case kempe showed temporal analogue problem temporal graphs concept literature also referred evolving graphs temporal networks multidimensional networks link streams networks work use model edge time stamp assuming discrete time steps equivalent sequence static graphs fixed set vertices formally define temporal graph follows definition temporal graph undirected temporal graph ordered triple consisting set vertices set maximal time label see figure example temporal graph four time steps also referred layers static graph obtained temporal graph removing time stamps call underlying graph many applications temporal graphs underlying mathematical models consider instance public transportation network edges correspond bus train connections stations vertices natural model connections time stamps examples include information spreading social networks communication social networks biological pathways spread diseases fundamental question temporal graphs addressing issues connectivity survivability robustness whether path distinguished start vertex distinguished target vertex thoroughly study computational complexity separating given temporal graph deleting vertices aim identify tractable cases end consider two similar still significantly differing path models used literature temporal graphs leading two corresponding models temporal separation sink usually denoted literature consistent michail use instead reserve refer points time table overview results herein abbreviates denote number vertices respectively refers underlying graph input temporal graph denotes fixed constant prop thm cor thm thm obs general temporal strict temporal planar temporal core general const const size open log log two models start introduction model given temporal graph two distinct vertices temporal length sequence vertex set temporal temporal state central problem paper temporal input temporal graph two distinct vertices question admit temporal size second model strict variant temporal called strict literature strict temporal paths also known journeys vertex set strict temporal strict temporal thus second main problem strict temporal defined complete analogy temporal replacing temporal separators strict ones strict variant appears natural variant viewed conservative version problem instance scenario spreading speed might unclear ensure containment spreading separating patient zero certain target temporal might safer choice contributions following summarize results see table survey section investigate computational hardness problems restrictions number layers restriction underlying graph planar prove temporal strict temporal respectively strengthening result kempe show moreover prove problems temporal graphs planar underlying graph end prove problem definition see section planar graphs section identify tractable cases problems temporal graphs layers problem hardness result already strict variant temporal equivalent static graphs identify dichotomy computational complexity proving solvability strict temporal side product give algorithm shortest strict temporal paths problem improving best known running time moreover prove constantly many layers strict temporal planar graphs solvable log time section introduce notion temporal cores temporal graphs informally temporal core temporal graph set vertices whose change time prove temporal graphs temporal core constant size temporal solvable log time fixed constant independent size temporal core strict temporal remains even temporal core empty particular virtue work results point choice model strict versus problem crucial impact computational complexity said problem noticeable since literature far used models without discussing subtle differences computational complexity related work main reference work kempe proved temporal contrast berman proved computing temporal edge deletion instead vertex deletion solvable context survivability temporal graphs liang modiano studied cuts edge deletion last consecutive time stamps moreover studied temporal maximum flow defined maximum number sets journeys two journeys set use temporal edge within time steps different notion temporal flows temporal graphs introduced akrida showed compute polynomial time maximum amount flow passing source vertex sink vertex given point time menger theorem states maximum number paths equals size static graphs menger theorem allows finding via maximum flow computations however berman proved analogue menger theorem temporal graphs asking maximum number strict temporal paths instead hold kempe proved former analogue menger theorem holds true underlying graph excludes fixed minor mertzios proved another analogue menger theorem maximum number strict temporal never leave vertex time equals minimum number node departure times needed separate node departure time vertex time point michail spirakis introduced famous traveling salesperson problem studied problem temporal graphs dynamic diameter informally speaking temporal graphs every two vertices reach time steps time erlebach studied problem temporal graphs underlying graph bounded degree bounded treewidth planar additionally introduced class temporal graphs regularly present edges temporal graphs edge associated two integers consecutive time steps absence edge axiotis fotakis studied problem finding smallest temporal subgraph temporal graph temporal connectivity preserved temporal graphs underlying graph bounded treewidth preliminaries convention denotes natural numbers without zero use static graphs use basic notations static graph theory let undirected simple graph use denote set vertices set edges respectively denote graph without vertices denotes induced subgraph path length sequence edges set path set vertices temporal graphs let temporal graph graph called layer temporal graph underlying graph temporal graph defined write short clear context define induced temporal subgraph say connected underlying graph connected let vertices visited denoted surveys concerning temporal graphs refer strict temporal separators throughout whole paper assume underlying graph temporal input graph connected furthermore accordance assume set ordered ascending labels moreover assume number layers number lemma let instance strict temporal separation algorithm computes time instance strict temporal equivalent observe layer temporal graph contains edge irrelevant temporal also holds true strict case hence delete layer temporal graph observation formalized following two data reduction rules reduction rule let temporal graph let interval layer edgeless graph replace reduction rule let temporal graph interval layer edgeless graph set prove next reduction rules exhaustively applicable linear time reduction rule applicable reduction rule applicable figure figure shows temporal graph reduction rule applicable particular layers edgeless figure shows temporal graph reduction rule applied exhaustively lemma reduction rules remove add temporal temporal graph exhaustively applied time proof first discuss reduction rule let temporal graph interval layer edgeless graph let temporal let graph applied reduction rule distinguish three cases case touched reduction rule hence also exists case temporal case clearly temporal direction works analogously look temporal compute corresponding temporal reduction rule exhaustively applied iterating set ordered ascending labels first given requirement appear set iterate replace next given requirement appear set iterate replace repeat procedure end reached since iterate done time reduction rule executed linear time iterating edges taking maximum label note sets remain untouched reduction rule hence application reduction rule add remove temporal consequence lemma maximum label number hence input size proof lemma let temporal graph reduction rules applicable thus regarding two models following connection lemma computable reduction strict temporal temporal maps instance instance proof let instance strict temporal construct equivalent instance lineartime set called set next let initially empty add completes construction note done time holds claim let temporal size claim also temporal suppose towards contradiction case temporal note vertices alternated vertices vertex corresponds edge temporal induced vertices contradiction observe temporal obtain temporal larger size contains vertices let temporal size containing vertices claim also temporal suppose towards contradiction case temporal path note obtain temporal adding consecutive vertices appears vertex contradiction hardness results section establish preliminary hardness results observe temporal strict temporal strongly related lbs problem given undirected graph two distinct vertices two integers subset length get following reduction observation reduction lbs strict temporal separation maps instance lbs instance strict temporal baier showed even lower bound path length five hence strict temporal hand lemma implies temporal however closer inspection get variant already improves previous result kempe showed temporal strict temporal summarize following proposition temporal strict temporal proof make use vertex cover problem figure vertex cover instance left corresponding temporal instance reduction proposition right dashed red solid green vertex gadgets dotted boxes vertex cover input graph question subset size holds let instance vertex cover say vertex cover size solution refine gadget baier theorem reduce vertex cover temporal let vertex cover instance construct temporal instance vertices defined note computed polynomial time vertex vertex gadget consists three vertices six addition edge edge gadget consists two see figure example prove let vertex cover size claim temporal vertices vertex cover exactly one vertex vertex vertex cover two vertices hence first consider vertex note two distinct temporal hence every temporal contains second let let temporal paths contain since vertex cover know least one element thus hence neither exist follows temporal size temporal let temporal size let recall two distinct temporal vertex gadget namely vertices vertex gadget hence form start preprocessing ensure vertex gadget one two separators let iterate case nothing case remove decrease one one observe temporal visiting still separated case remove add one observe still temporal size case remove add one observe still temporal size complete case distinction neither separate temporal vertex gadget construct vertex cover taking since vertex gadgets containing either one two vertices follows assume towards contradiction vertex cover edge hence contradicts fact temporal temporal follows vertex cover size next section prove bound tight strict case note case tightness obvious first case observe significant difference strict variant separation problem since respect postulated separator size observation lemma obtain following corollary temporal strict temporal respect furthermore show temporal strict temporal restricted class planar temporal graphs temporal graphs planar underlying graph end prove planar result consider independent interest note planar graphs known variant lbs undirected graphs weighted directed graphs theorem planar graphs figure planar graph left edge costs edges obtained reduction theorem connector sets highlighted gray indicated dashdotted lines roughly idea behind theorem reduce planar variant constant vertex degree since degree constant one replace vertex gadget proof give reduction variant lengthbounded referred planar input graph planar edge costs maximum degree degree three incident outerface let instance planar assume construct instance follows construction vertex introduce grid size graph vertex set uvi edge set uvi six pairwise disjoint subsets size refer connector sets fix orientation two connector sets top two bottom one left one right formally note length vertices let plane embedding say edge incident vertex position ith edge incident counted clockwise respect edge introduce differs weight follows let position position constructed follows add path consisting vertices connect odd since incident outerface add path length endpoints set budget edge deletions one endpoint vertex cvi edge connect endpoint vertex cwj edge constructed follows introduce planar matching vertices cvi cwj instance connect vertex connect vertex omit remaining cases replace edge path length least endpoints identified endpoints replaced edge hence path two length least next choose connector sets csi czj vertex csi czj adjacent vertex always exist degree three add two special vertices edges vertex csi well vertex czj finally set note computed polynomial time moreover one observe planar obtaining embedding correctness claim let thus solution length construct set size taking one arbitrary vertex note since edge cost one assume towards contradiction shortest length since path two length least know goes otherwise would length least reconstruct corresponding taking edge goes hence length contradicts length consequently length let thus solution minimum size length since minimum size follows following claim claim let vertex set size holds cvi cvj length proof claim let cvi cvj two connector sets vertices edges cvi cvj different cases difficult see case claim follows menger theorem note minimality holds construct edge set cost taking assume towards contradiction shortest length reconstruct corresponds follows first take edge csi always exists let first edge position add csi due claim always exists length take recall path length due claim connect two consecutive edges path length let last edge position czj czj add length claim note visits length contradicts forms solution follows length observation lemma planarity preserving get following corollary temporal strict temporal planar temporal graphs temporal graphs layers consider situation commuter wants reach working place home public transport certain time needs work certain time span willing spend traveling home work hence considering available transportation network certain time interval relevant restrict transportation network time interval reasonable assume layers present scenario temporal size commuter reach working place time even transport hubs blocked previous section showed strict temporal already five two layers proposition section determine smaller number layers whether respective problems solvable folklore minimum static graphs computed polynomial time thus temporal solvable strict variant interesting show section dichotomy takes place increases four five solve strict temporal time furthermore show strict temporal planar temporal graphs solvable log time arbitrary fixed number layers studying planar temporal graphs instance context transportation since many street networks modeled planar graphs strict static expansion key tool version temporal graph reduces reachability related questions temporal graphs similar questions directed graphs introduce similar tool strict temporal paths let temporal graph let define sets strict static expansion directed acyclic graph acol referred acol figure temporal graph left strict static expansion right one strict temporal corresponding marked green observe strict temporal correspondence refer figure example lemma let temporal graph two distinct vertices strict static expansion computed time proof let temporal graph two distinct vertices note connected construct strict static expansion four steps follows first initiate empty linked list second iterate set labels add already exist added vertex push linked list add done time observe two consecutive entries linked list third iterate linked list increasing order add entry note sizes linked lists sum last add well edges note well employed constant number procedures running time thus computed time shortest strict temporal paths say strict temporal shortest strict temporal length subroutine hidden algorithmic results need solve shortest strict temporal paths problem temporal graphs find shortest strict paths source vertex vertices temporal graph indeed provide algorithm believe independent interest improves adaptations model details see section previous results contrast algorithm subroutine adjusted case proposition shortest strict temporal paths solvable time number high level algorithm proposition constructs directed acyclic graph temporal graph solves shortest paths constructed directed acyclic graph linear time see cormen section following proof makes use strict static expansion temporal graph proof lemma compute strict static expansion time define weight function otherwise observe weighted directed acyclic graph weight equal length corresponding strict temporal hence use algorithm makes use topological order compute shortest time cormen section iterate construct shortest strict temporal shortest done time consequently overall running time since shortest strict temporal length algorithm asymptotically optimal adaptations proposition model considered model temporal graph directed traversal time context strict temporal path always one excluded case pointed algorithms adjusted allow however possible algorithm strict static expansion contain cycles hence assume directed let directed temporal graph denote directed layer first initiate many linked lists without loss generality assume see lemma second construct directed temporal graph empty beginning iterate set ascending labels add add new vertex add linked list call original edge connector edge reach directed label first time add directed linked list observe strict temporal corresponding strict temporal additionally ordered ascending labels constructed time construct strict static expansion directed temporal graph modify edge set finally adjust weight function algorithm proposition columnedges correspond otherwise observe strict temporal traversal time corresponding strict static expansion weight traversal time temporal graphs layers ready give algorithm strict temporal temporal graphs theorem strict temporal maximum label solved time algorithm executes following steps preprocessing step remove unnecessary vertices graph compute auxiliary graph called directed path cover graph temporal graph compute separator directed path cover graph following explain steps detail preprocessing reduces temporal graph following properties definition temporal graph two distinct vertices reduced strict temporal contains strict temporal length two preprocessing step performed polynomial time lemma let instance strict temporal time one either decide construct instance strict temporal equivalent reduced following proof makes use strict static expansion temporal graph see section details proof first remove used strict temporal let instance strict temporal execute following procedure construct strict static expansion perform search mark vertices search tree reachable let reachable vertices iii construct observe reachable part directed arcs change direction instance perform search mark vertices search tree reachable let reachable set vertices graph vertices reachable vertex vertex reachable output temporal graph one observe temporal subgraph note subroutines computable time see lemma consequently computed time claim strict temporal contains let assume towards contradiction strict temporal contains construction know hence contains either thus corresponding strict temporal well contains contradiction figure left side depicts excerpt reduced temporal graph maximum label dashed arcs labeled number indicate shortest strict temporal path length right side depicts directed path cover graph reduced temporal graph gray arc vertex set vertex set denotes two vertices arc take example vertex vertex furthermore claim strict temporal strict temporal let strict temporal since strict temporal corresponding note witness vertices well follows construction also strict temporal direction works analogously let strict temporal length two one observe strict temporal must contain proposition compute shortest strict temporal time check whether length two case remove vertex decrease one note find strict temporal length two done time decided yet whether construct strict temporal instance minus number strict temporal length two found finally observe reduced removed vertices thus showed minimum size paths length four graph equal number length four graph adjust idea strict temporal paths temporal graphs proof implicitly relies transitivity connectivity static graphs hold temporal graphs hence put extra effort adapt result temporal case end define directed auxiliary graph definition directed path cover graph let reduced temporal graph two distinct vertices directed path cover graph directed graph iii herein vertex set shortest strict temporal length shortest strict temporal length figure depicts generic directed path cover graph reduced temporal graph note definition reduced temporal graphs set always empty hence depicted crucial property allows prove following lemma let reduced temporal graph let two distinct vertices directed path cover graph computed time strict temporal introduce notation use following proof departure time arrival time temporal traversal time length proof let reduced temporal graph two distinct vertices need know order compute directed path cover graph shortest strict temporal shortest strict temporal done time proposition iterate set hence directed check whether arc path cover graph computed say strict temporal length chordless contain strict temporal length exists call chord obviously set vertices contain chordless strict temporal contain strict temporal discuss appearance directed path cover graph assume towards contradiction vertex since reduced strict temporal length two thus vertex must distinguish two cases case let since reduced know strict temporal contains strict temporal must length least three reduced observe length strict temporal lower bound arrival time hence arrival time least three contradiction last therefore equal arrival time consequently case let since reduced know strict temporal contains strict temporal must length least three reduced observe first hence departure time least three since length least three arrival time least five contradiction maximum label four consequently observe sets partition strict temporal length four figure see appearance directed path cover graph claim chordless strict temporal let chordless strict temporal since reduced length three four case let length three visited since strict temporal length two forces easy see existence existence easy see need strict temporal length one assume towards contradiction would strict temporal length one words labels least three thus least three otherwise strict temporal length two must strict temporal uses otherwise reduced furthermore must length least three otherwise reduced thus departure time two first incident hence contradicts existence strict temporal length one hence definition know case let length four thus looks follows immediately follows shortest strict temporal length one furthermore shortest strict temporal length two otherwise would chordless claim assume towards contradiction observe shortest strict temporal length one thus must part strict temporal reduced since note would strict temporal length two hence definition know let know also strict temporal chordless strict temporal let strict temporal remains shown assume towards contradiction thus path length either three four see figure case let length three visited thus see figure definition know note must part strict temporal strict temporal length two hence since strict temporal assumption either case let must part strict temporal therefore either exist hence strict temporal holds contradicts existence case let strict temporal contains timeedge since either exist hence strict temporal contradicts existence case let length four visited visited considering figure one observe definition know strict temporal containing reduced since know length four first two assume towards contradiction visits hence third first two strict temporal length two arrival time least two thus observe strict temporal path length one hence contradicts visiting consequently strict temporal visits second hence implies vertex see figure strict temporal contains reduced since strict temporal strict temporal length least two thus also arrival time least two hence strict temporal contains either exist otherwise would strict temporal length one implies hence sequence forms strict temporal contradicts existence summary exist consequently figure shows construct reduced temporal graph set empty indicates algorithm fails finally lemmata prove theorem proof theorem let instance strict temporal first apply lemma time time either decide obtain instance strict temporal second case compute directed path cover graph time lemma next check whether size time figure reduced temporal graph maximum label vertex set directed path cover graph empty red green edges strict temporal paths show edges removed graph reduced definition furthermore removed since strict temporal path folklore result lemma size strict temporal size since lemma reduced overall running time planar temporal graphs employ optimization variant courcelle theorem show strict temporal planar temporal graphs efficiently solvable maximum label constant theorem strict temporal planar temporal graphs solved log time maximum label constant prove theorem first introduce mso state optimization variant courcelle theorem monadic mso logic consists countably infinite set individual variables unary relation variables vocabulary formula monadic logic graphs constructed vertex variables edge variables incidence relation edge vertex variables inc relations quantifiers make use many folklore shortcuts example check whether vertex variable check whether edge variable treewidth graph measures graph paper using following theorem treewidth graph black box formal definitions treewidth mso refer courcelle engelfriet theorem arnborg courcelle engelfriet exists algorithm given mso formula free monadic variables affine function iii graph finds minimum maximum evaluations formula satisfied running time length treewidth proof theorem let instance strict temporal separation underlying graph planar define optimization variant strict temporal monadic second order logic mso show sufficient solve strict temporal temporal graph treewidth underlying graph function depending define graph underlying graph labeling otherwise observe binary representation bit one exists time point compute first iterate set add edge edge set underlying graph whenever find furthermore associate edge initially zero gets increment appropriate data structure done log time observe end finally copy vertex set hence compute log log time arnborg showed possible apply theorem graphs edges labels fixed finite set either augmenting graph logic incorporate predicates describing labels representing labels unquantified edge set variables since theorem gives algorithm respect input size conclude size label set depends theorem still gives algorithm arbitrary fixed size mso formula treewidth define optimization variant strict mso temporal first mso formula layer checks whether edge present layer bit note length layer function second write mso formula tempadj inc inc layer determine whether two vertices adjacent time point since length layer function length tempadj function well third mso formula path tempadj check whether strict temporal visit vertex observe length path function finally theorem solve optimization variant strict temporal formula affine function time computable function gives overall running time log decide strict temporal note every strict temporal induced length hence remove vertices reachable path length find set vertices reachable path length breadth first search linear time therefore sufficient solve strict temporal observe diameter planar treewidth planar graph three times diameter see flum grohe thus optimization variant strict temporal solved log time computable function constant yields running time log temporal graphs small temporal cores several scenarios reasonable assumption instance strict temporal small number vertices adjacent temporal edges edges appear layers consider instance following setting employee university berlin uses bike every day ride apartment university campus cover large distances possibility take bike onto city train transportation network consists potentially large number traffic nodes connected bike lanes static edges appear every layer manageable number city railway station potentially connected bike lanes railway lines temporal edges city trains always available temporal size exist employee reach university bike even traffic hubs blocked section investigate complexity computing strict temporal temporal graphs number vertices adjacent temporal edges small call set vertices temporal core temporal graph definition temporaltcore temporal graph vertex set called temporal core strict temporal observe hardness reduction described observation produces instance empty temporal core hence get following result observation strict temporal stark contrast temporal temporal graphs temporal core solved polynomial running time degree polynomial depend size temporal core formally show following theorem temporal solvable log log time proof solve instance temporal using temporal core use algorithm cygan node multiway cut see step recall definition node multiway cut node multiway cut input undirected graph set terminal integer question set size every distinct remark cygan algorithm modified obtain solution let instance temporal temporal given without loss generality assume otherwise add two vertices one incident layer one incident temporal layer one furthermore need notion maximal static subgraph graph contains edges appear every layer specifically algorithm works follows note compute temporal core log time figure illustration idea behind proof theorem side sketch temporal graph enclosed ellipse temporal separator red hatched induced partition outer rings contain open neighborhood sets side sketch constructed graph enclosed ellipse partition guessed steps nodes edges neighborhood respectively created step guess set size guess number partition adding vertex part add edge construct graph copying sets ngb add edge ngb solve node multiway cut instance solution found solution output yes possible guesses considered without finding solution output see figure visualization constructed graph since sanity check step suffices show temporal size partition different parts node multiway cut instance solution size solution temporal correctness let temporal size first set let connected components construct partition easy see different parts partition observe hence vertices different connected components different connected components thus solution size node multiway cut instance proving correctness remains prove let solution size node multiway cut instance need prove forms temporal clearly done arguments thus assume since solution know different connected components hence vertices different connected components assume towards contradiction temporal observe hence two different vertices temporal vertices visited hence contained note disconnected since visit vertices hence conclude connected connected contradicts fact solution running time remains show algorithm runs proposed time guess step many possibilities guess step many possibilities bell function step sanity check step clearly done polynomial time let minimum temporal size otherwise cygan showed node multiway cut solved time min since know hence step done time thus overall running time log log conclude strict variant temporal behave differently temporal graphs temporal core strict version stays version becomes solvable concluding discussion outlook one general conclusion informally speaking temporal path model strongly matters assessing computational complexity finding small separators temporal graphs phenomenon far neglected literature showed temporal strict temporal already temporal graphs respectively completing picture showed problems respective constants tight problem becomes solvable number layers respective constant showed strict temporal temporal graphs planar underlying graphs remains however proved graph class additionally number layers constant strict temporal solvable log time leave open whether temporal separation admits similar statement finally introducing temporal core notion proved temporal solvable polynomial time temporal graphs core discuss two research directions consider particularly interesting temporal graph classes initiated study strict temporal planar temporal graphs future research could include continuation study problems planar temporal graphs parameterized complexity approximation point view point think planar temporal graphs particular interest underlying graphs several temporal graphs instance commuting traffic street public transport networks planar almost planar furthermore notions temporal graph classes could practical theoretical interest future research may include study specific temporal graph classes including viewpoint parameterized complexity temporal graph parameters second natural direction study structural parameters temporal graphs instance introduced studied notion temporal core small temporal core allows efficiently solving temporal one may ask structural parameters temporal graphs allow tractability references akrida mertzios spirakis temporally connected graphs small cost proceedings international workshop approximation online algorithms waoa pages springer akrida czyzowicz kuszner spirakis temporal flows temporal networks proceedings international conference algorithms complexity ciac pages springer arnborg lagergren seese easy problems graphs journal algorithms axiotis fotakis size approximability minimum temporally connected subgraphs proceedings international colloquium automata languages programming icalp volume lipics pages schloss dagstuhl fuer informatik baier erlebach hall kolman schilling skutella cuts flows acm transactions algorithms berman vulnerability scheduled networks generalization menger theorem networks boccaletti bianconi criado del genio romance wang zanin structure dynamics multilayer networks physics reports casteigts flocchini quattrociocchi santoro graphs dynamic networks international journal parallel emergent distributed systems corley sha vital links nodes weighted networks operations research letters cormen leiserson rivest stein introduction algorithms edition mit press courcelle engelfriet graph structure monadic logic languagetheoretic approach volume cambridge university press cygan pilipczuk pilipczuk wojtaszczyk multiway cut parameterized lower bounds acm trans comput theory diestel graph theory edition volume graduate texts mathematics springer erlebach hoffmann kammer temporal graph exploration proceedings international colloquium automata languages programming icalp pages springer ferreira building reference combinatorial model manets ieee network flum grohe parameterized complexity theory texts theoretical computer science eatcs series springer verlag berlin fluschnik hermelin nichterlein niedermeier fractals kernelization lower bounds application cut problems proceedings international colloquium automata languages programming icalp volume pages schloss dagstuhl fuer informatik full version https ford fulkerson maximal flow network canadian journal mathematics golovach thilikos paths bounded length cuts parameterized complexity algorithms discrete optimization himmel molter niedermeier sorge adapting algorithm enumerating maximal cliques temporal graphs social network analysis mining holme modern temporal network theory colloquium european physical journal holme temporal networks physics reports kempe kleinberg kumar connectivity inference problems temporal networks journal computer system sciences liang modiano survivability networks ieee transactions mobile computing plummer mengerian theorems paths bounded length periodica mathematica hungarica menger zur allgemeinen kurventheorie fundamenta mathematicae mertzios michail chatzigiannakis spirakis temporal network optimization subject connectivity constraints proceedings international colloquium automata languages programming icalp pages springer michail introduction temporal graphs algorithmic perspective internet mathematics michail spirakis traveling salesman problems temporal graphs theoretical computer science pan schild interdiction problems planar graphs discrete applied mathematics scellato leontiadis mascolo basu zafer evaluating temporal robustness mobile networks ieee trans mob schieber khuller complexity finding vital arcs nodes technical report college park usa viard latapy magnien computing maximal cliques link streams theoretical computer science cheng huang huang efficient algorithms temporal path computation ieee transactions knowledge data engineering
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inner amenability property gamma mcduff factors stable equivalence relations tobe stefaan nov abstract say countable group mcduff admits free ergodic probability measure preserving action crossed product mcduff factor similarly said stable admits action orbit equivalence relation stable mcduff property stability inner amenability property gamma subtly related several implications non implications obtained complete picture remaining implications counterexamples introduction murray von neumann proved hyperfinite factor isomorphic free group factor showing property gamma recall tracial von neumann algebra property gamma exists net unitariespun weakly limn kxun say countable group property gamma shown property gamma inner amenable exists conjugation invariant mean every finite subset converse hold inner amenable group infinite conjugacy classes icc property gamma constructed factor admits non commuting central sequences precisely central sequence algebra non abelian said mcduff see counterpart mcduff property equivalence relations introduced called stability countable ergodic probability measure preserving pmp equivalence relation called stable unique hyperfinite ergodic pmp equivalence relation theorem stability characterized terms central sequences clearly stable factor mcduff mcduff property stability closely related inner amenability countable group free ergodic pmp action consider group measure space factor orbit equivalence relation stable group inner amenable actually already mcduff must inner amenable see proposition say group stable resp mcduff admits free ergodic pmp action orbit equivalence relation stable resp crossed product mcduff follows probability spaces assumed standard thus following subtly related notions countable group inner amenability property gamma mcduff property stability figure summarizes implications leuven department mathematics leuven belgium supported phd fellowship research foundation flanders fwo leuven department mathematics leuven belgium supported part european research council consolidator grant long term structural funding methusalem grant flemish government non implications properties main contribution article fill arrows unknown far prove kida counterexamples provide icc groups property gamma mcduff also adapt example necessarily non icc property yet mcduff group obtain icc mcduff group stable property gamma thm inner amenable thm prop mcduff stable thm figure implications non implications countable discrete icc group dashed arrows proven article later use recall theorem subgroup countable group relative property every unitary representation every bounded sequence vectors satisfying limn limn kph denotes orthogonal projection onto closed subspace vectors context tracial von neumann algebras characterization relative property gives following well known lemma completeness include proof lemma let tracial von neumann algebra let countable group subgroup let homomorphism satisfying denote preserving conditional expectation von neumann subalgebra relative property define unique trace bounded sequence satisfying limn limn kan proof suffices observe defines unitary representation orthogonal projection onto subspace vectors icc group property gamma mcduff kida constructed first icc groups property gamma particular inner amenable stable meaning admit stable free ergodic pmp action prove groups even mcduff meaning admit free ergodic pmp action crossed product mcduff fix property group admits central element infinite one could take define given take order given two non zero integers define hnn extension theorem kida group defined icc property gamma mcduff proof analyzing reduced words hnn extensions easy check icc relatively icc subgroup meaning infinite every see ozawa already proved property gamma proof found theorem completeness include following slightly easier proof claim exists sequence borel functions satisfying lim lim denote canonical unitary elements given sequence functions denoting fourier coefficients define unitary elements given upk says since moreover follows non trivial central sequence property gamma prove existence satisfying define semidirect product abelian group acting automorphisms given multiplication power define action given equip lebesgue measure class since amenable group follows orbit equivalence relation hyperfinite thus strongly ergodic provides sequence unitary elements tending weakly approximately sense strongly restricting interval extending periodically find sequence satisfying choose free ergodic pmp action prove factor mcduff let central sequence put since property lemma kun unique trace preserving conditional expectation since relatively icc follows also define since kun kun get kun since conclude kun define denotes von neumann algebra functions claim subalgebras form commuting square prove claim suffices take prove let fourier decomposition since commutes get particular since claim follows since kun kun conclude kun abelian mcduff icc mcduff group stable two obvious obstructions stability group non inner amenability property theorem using example mcduff property group constructed particular example mcduff group stable however theorem mcduff property group never icc section construct examples icc mcduff groups stable giving following obstruction stability combining methods proposition countable group admits subgroup relative property relative icc property stable proof let free ergodic pmp action denote associated orbit equivalence relation write denote normalizer inside prove stable theorem show central sequences projection satisfy kun put also define algebra functions denote unique trace preserving conditional expectations since relative property lemma get kun kpn since relatively icc therefore commute follows kun every countable group denote defined normal subgroup consisting elements finite conjugacy class also recall group said virtually abelian admits abelian subgroup finite index denote centralizer subset construct icc mcduff group stable start property group theorem residually finite property group virtually abelian proved group indeed exists theorem know mcduff perform iterative amalgamated free product construction order embed relatively icc subgroup larger group way remains mcduff fix decreasing sequence finite index normal subgroups enumerate note every finite index subgroup decreasing sequence finite index normal subgroups put define inductively amalgamated free product view define direct limit theorem group constructed icc mcduff property stable proof first prove relatively icc take take since infinite amenable property infinite index infinite index follows infinite set prove icc suffices prove every infinite conjugacy class take let generator description since elements distinct since property follows proposition stable remains construct free ergodic pmp action mcduff every define homomorphism given identity map mapping writing obtain compatible system homomorphisms combine homomorphism particular induction one checks normal subgroup commutes every denote profinite action given inverse limit define note ergodic pmp action every consider bernoulli action equipped product measure define note weakly mixing pmp action define product define diagonal action since weakly mixing ergodic diagonal action ergodic take since bernoulli action essentially free set measure zero thus also measure zero conclude essentially free denote let tracial state prove mcduff use method theorem denote natural induced coordinate maps definition implies every large enough every define element whenever direct computation shows every homomorphism commutes follows commutes whenever also get commutes commutes choosing sequence get vhn central sequence unitaries vhn since finite index subgroup know non abelian thus choose sequences corresponding central sequences vhn satisfy vhn mcduff factor mcduff groups inner amenable icc groups following result consequence main theorem non icc case need small extra argument follow convention group inner amenable exists conjugation invariant mean gives weight zero finite subsets subset proposition every mcduff group inner amenable proof let group inner amenable let free ergodic pmp action denote let arbitrary central sequence prove limn kxn write every fix since limn limn let maximal projection satisfying put since action essentially free get follows limn kpi every limn kan define unit vectors given kan previous paragraph proved limn every vectors approximately conjugation invariant since inner amenable functions must satisfy limn meaning limn kxn proved every central sequence satisfies limn kxn follows abelian mcduff references choda inner amenability fullness proc amer math soc connes feldman weiss amenable equivalence relation generated single transformation ergodic theory dynam systems cornulier finitely presentable groups kazhdan property infinite outer automorphism group proc amer math soc effros property inner amenability proc amer math soc ershov kazhdan groups whose virtually abelian preprint available http jolissaint property pairs topological groups enseign math jones schmidt asymptotically invariant sequences approximate finiteness amer math kida inner amenable groups stable action geom dedicata kida stability orbit equivalence groups vaes groups groups geom dyn mcduff central sequences hyperfinite factor proc london math soc murray von neumann rings operators ann math popa vaes fundamental group factors equivalence relations arising group actions quanta maths proceedings conference honor connes birthday clay mathematics institute proceedings stalder extensions hnn ann inst fourier grenoble vaes inner amenable group whose von neumann algebra property gamma acta math
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robust attitude tracking control aerobatic helicopters geometric backstepping approach nidhish ravi mangal department aerospace engineering iit kanpur kanpur india systems control group iit bombay mumbai india sep abstract robust attitude tracking control aerobatic helicopter using geometric backstepping techniques presented article nonlinear coupled dynamics model helicopter considered wherein rotor flap dynamics modeled first order system fuselage rigid body dynamically coupled rotor system robustness controller presence structured unstructured disturbances explored structured disturbance due uncertainty rotor parameters unstructured perturbation modeled exogenous torque acting fuselage performance controller demonstrated presence types disturbances simulations unmanned helicopter work possibly first systematic attempt designing globally defined robust attitude tracking controller aerobatic helicopter retains rotor dynamics incorporates uncertainties involved introduction conventional helicopter unmanned aerial vehicles uavs used wide variety missions ranging surveillance precision agriculture movie making pipeline inspection real estate wild life monitoring etc name helicopters single main rotor tail rotor capable performing extreme aerobatic maneuvers gavrilets frazzoli mettler piedmonte feron abbeel coates gerig maneuvers may critical towards survivability vehicle extreme circumstances presence strong flight cluttered congested environment combat scenarios aggressive maneuvers involve large angle rotations high angular velocity inverted flight pirouette etc necessitates tracking controller globally defined capable achieving aggressive rotational maneuvers attitude tracking problem helicopter significantly different rigid body smallscale helicopter modeled coupled interconnected system consisting fuselage rotor control moments generated rotor excite rigid body dynamics fuselage affects rotor loads dynamics causing nonlinear coupling key differences rigid body tracking problem attitude tracking helicopter following presence large aerodynamic damping rotational dynamics required control moment tracking applied instantaneously due rotor blade dynamics control moments produced rotor subsystem inherently first order dynamics mettler significance including rotor dynamics controller design helicopters extensively studied literature hall bryson takahashi ingle celi panza lovera hall bryson shown importance rotor state feedback achieving tight attitude control large scale helicopters takahashi compares attitude controller design cases without rotor state feedback similar work ingle celi investigated effect including rotor dynamics various controllers namely lqg eigenstructure assignment meeting stringent handling quality requirements conclude controllers designed meet high bandwidth requirements rotor dynamics robust required lower control activity ones designed without including rotor dynamics panza lovera used rotor state feedback designed controller robust also fault tolerant respect failure rotor state sensor previous attempts helicopter attitude control mostly based attitude parametrization euler angles suffer singularity issues quaternions email addresses nraj nidhish raj ravi banavar abhish abhishek mangal mangal kothari doctoral student iit kanpur india professor systems control engineering iit bombay india assistant professor department aerospace engineering iit kanpur india ambiguity representation tang yang qian zheng explicitly consider rotor dynamics design stabilizing controller based sliding mode technique using euler angles hence confined small angle maneuvers raptis valavanis moreno designed position tracking controller smallscale helicopter wherein inner loop attitude controller based rotation matrix consider rotor dynamics marconi naldi designed position tracking controller flybared stabilizer bar miniature helicopter robust respect large variations parameters made simplifying assumption disregarding rotor dynamics taking static relation flap angles cyclic input stressing significance rotor dynamics ahmed pota ahmed pota garratt developed backstepping based stabilizing controller using euler angles smallscale flybared helicopter inclusion servo rotor dynamics provided correction terms controller incorporate effect servo rotor dynamics near hover conditions zhu huo developed robust nonlinear controller disregarding flap dynamics frazzoli dahleh feron developed coordinate chart independent trajectory tracking controller configuration manifold helicopter flap dynamics taken account synthesis robust attitude tracking control flybarless aerobatic helicopter presented paper coupled model helicopter wherein fuselage modeled rigid body rotor first order system considered disturbances added fuselage rotor dynamics disturbance added fuselage dynamics unstructured captures effect exogenous torque acting helicopter lumped together torque could arise due center mass misalignment main rotor shaft axis moment due tethered load attached point different center mass effect fuselage hand rotor dynamics structured disturbance form uncertainties parameters considered parameters rotor dynamics rotor stiffness time constants estimated flight data means system identification hence prone erroneous discussed paper controller proposed raj banavar abhishek kothari tracking completely fails using overestimated value main rotor time constant controller paper robust attitude tracking controller designed using notions geometric control backstepping approach renders solutions associated error dynamics uniformly ultimately bounded observed numerical simulations proposed controller capable performing aggressive rotational maneuvers presence aforementioned structured unstructured disturbances paper organized follows section describes dynamics helicopter explains effect aerodynamic damping motivates need robust attitude tracking controller section presents proposed robust attitude tracking controller helicopter dynamics efficacy proposed controller demonstrated numerical simulation section helicopter model unlike quadrotors helicopter modeled rigid body capture dynamics required high bandwidth attitude control purposes hence consider hybrid model small scale flybarless helicopter consists fuselage rotor fuselage modeled rigid body rotor first order system generates required control moment inclusion rotor model crucial since introduces significant aerodynamic damping system integral part dynamics helicopter clearly distinguishes helicopter control problem rigid bodies space robotics applications interplay present rotor tip path plane hub plane fuselage figure fuselage tip path plane rotational equations motion fuselage given rotation matrix transforms vectors body fixed frame inertial frame reference external moment acting fuselage due rotor disturbance torque bounded body moment inertia fuselage angular velocity fuselage expressed body frame lie algebra isomorphism given hat operator first order tip path plane tpp equations rotor considered capture required dynamics gross movement fuselage mettler coupled flap equation rotor given mettler chen respectively longitudinal lateral tilt rotor disc respect hub plane shown fig main rotor time constant control inputs rotor subsystem respectively lateral longitudinal cyclic blade pitch angles actuated servos swash plate mechanism blade root stiffness main rotor angular velocity blade moment inertia flap hinge equation introduces cross coupling flap angle angular velocity rewritten note effect angular velocity cross coupling effectively canceled using fuselage angular velocity feedback since rotor angular velocity measured accurately using autopilot rotor disc hub plane fuselage figure coupling coupling rotor fuselage occurs rotor hub rolling moment pitching moment acting fuselage due rotor flapping consists two components due tilting thrust vector due rotor hub stiffness given distance rotor hub center mass helicopters rotor hub stiffness much larger component due tilting thrust vector see table thus nominal variation thrust would result small variation equivalent hub stiffness control moment yaw axis applied tail rotor due higher rpm much faster aerodynamic response main rotor flap dynamics tail rotor along actuating servo modeled first order system tail rotor time constant main rotor dynamics tail rotor dynamics could written terms control moments input symmetric skew symmetric parts denoted respectively note combined dynamics given given form simple mechanical system bullo lewis actuator dynamics first order precludes equations motion written usual form geodesic riemannian manifold aerodynamic damping comes presence negative angular velocity terms positive angular velocity builds negative flap angles leading negative moment vice versa effect damping attitude dynamics helicopter parameters table depicted fig initial angular velocity roll axis damped zero less second maximum damping moment generated process significant torque rigid body size aerobatic maneuvers involve attitude trajectories large angular velocities therefore need design controllers cancel damping effect effect damping canceled effectively augmenting control input angular velocity feedback exact cancellation requires knowledge true value main rotor time constant overestimate used canceling would result positive feedback angular velocity could make attitude dynamics unstable shown fig restrict analysis case rotor stiffness parameters known perfectly although easily extended include uncertainties assumption justified observed numerical simulations even significant error lead tracking instability motivates attempt made paper make attitude controller robust respect uncertainty main rotor time constant damping moment time time damping moment generated angular velocity figure effect aerodynamic damping attitude dynamics helicopter response initial condition roll axis estimates rotor time constants given uncertainty parameters satisfy max max define maximum variation parameters max max max estimates given using relations max robust attitude tracking helicopter given twice differentiable attitude reference command objective design attitude tracking controller helicopter combined dynamics reproduced convenience order apply backstepping approach first tracking problem transformed stabilization error dynamics robust attitude tracking controller fuselage subsystem designed described lee leok harris mcclamroch subsequently extended include rotor dynamics configuration space fuselage subsystem lie group pointed maithripala berg dayawansa general tracking problem lie group reduced configuration stabilization problem identity element group possible lie group since error two configurations naturally defined using group operation operation always defined general configuration manifold set rotation matrices rotation error matrix current rotation desired rotation defined transforms vector current body frame desired body frame obtain fuselage error dynamics differentiate defining reduces differentiate obtain fuselage error dynamics defining difference actual desired moment acting fuselage due rotor differentiating gives rotor error dynamics amd equilibrium error dynamics corresponding zero tracking error hence meant stabilized tracking error stabilizing controller proportional derivative plus components proportional action derived tracking error function defined single critical point within sub level set identity sublevel set represents set rotations less radians identity derivative given rotation error vector derivation uses fact inverse hat map trace product symmetric skew symmetric matrices zero total derivative since positive definite quadratic within sub level set positive makes uniformly quadratic identity bullo lewis implies exists positive constants lee ker ker following definition ultimate boundedness taken khalil given sake completeness definition consider system piecewise continuous locally lipschitz contains origin equilibrium solutions uniformly ultimately bounded ultimate bound exist positive constants independent every independent following lemma uses lyapunov analysis show ultimate boundedness variation theorem khalil lemma let domain contains origin continuously differentiable function positive consider sublevel set every initial condition solution uniformly ultimately bounded ultimate bound exists proof see appendix following theorem presents main result paper theorem initial conditions starting set positive solutions error dynamics rendered uniformly ultimately bounded following choice control input ultimate bound given matrices given proof consider following positive definite quadratic function sublevel set time derivative function made negative definite setting virtual control input change variable would make ker error dynamics given candidate lyapunov function fuselage dynamics given given ker setting equation would result ker ker ker ker adding subtracting would give ker ker augmenting lyapunov function fuselage quadratic form gives candidate lyapunov function complete dynamics derivative bounded ker amd since symmetric part inequality written ker amd setting would make inequality ker given respectively consider last term inequality max setting using relation would result since positive definite therefore next ultimate boundedness tracking error dynamics shown positive definite quadratic positive guaranteed result satisfies following inequality sublevel set positive definite given along solution error dynamics guaranteed negative definite superlevel set condition lemma satisfied set met therefore follows lemma solutions error dynamics uniformly ultimately bounded ultimate bound given remark could independently set based uncertainties associated fuselage rotor dynamics important design flexibility helicopter since allows adjusting robustness controller exogenous torque independent uncertainties rotor parameters exogenous torque depends type mission helicopter flies externally attached payload cable suspended load rotor parameters remain constant given rotor hub blade properties remark proposed controller requires flap angle feedback case large scale helicopter easily measured described kufeld balough cross studebaker small scale helicopter instrumentation required flap angle measurement challenging rotor hingeless flap however observer flap angle implemented assumption remaining states available attitude angular velocity independently estimated using onboard inertial measurement unit proposed mahony hamel pflimlin simulation results tracking controller given simulated class model helicopter whose parameters given table helicopter given initial attitude deg pitch angle subjected sinusoidal roll angle input amplitude twenty degree frequency hertz simulations performed separately structured unstructured combined structured plus unstructured uncertainties case structured uncertainty main rotor time constant used controller implementation assumed percent true value tracking error dynamics unstable nominal controller without robustification term used shown fig hand robust controller performs satisfactorily evident fig unstructured disturbance introduced adding disturbance torque form sin cos case performance controller without robustification term compared figs figure shows performance proposed robust controller combined structured unstructured disturbances following controller parameters used simulation parameter jzz values table helicopter parameters roll angle deg ref roll rate roll rate response pitch rate response ref roll command roll pitch roll rate tracking response roll tracking response cyclic angle deg flap angle deg time time time time cyclic angle flap angle figure nominal controller response structured uncertainty ref roll command roll pitch roll angle deg ref roll rate roll rate response pitch rate response time roll tracking response roll rate tracking response cyclic angle deg flap angle deg time time time flap angle cyclic angle figure robust controller response structured uncertainty ref roll command roll pitch roll angle deg ref roll rate roll rate response pitch rate response time roll tracking response roll rate tracking response time cyclic angle deg flap angle deg time time flap angle cyclic angle figure nominal controller response unstructured uncertainty ref roll command roll pitch roll angle deg ref roll rate roll rate response pitch rate response time roll tracking response roll rate tracking response cyclic angle deg flap angle deg time time time cyclic angle flap angle figure robust controller response unstructured uncertainty ref roll command roll pitch roll angle deg ref roll rate roll rate response pitch rate response time time roll tracking response roll rate tracking response cyclic angle deg flap angle deg time time cyclic angle flap angle figure robust controller response structured unstructured uncertainty conclusion paper classifies possible uncertainties associated model helicopter structured unstructured disturbances proposed controller robust respect uncertainties tracking error shown ultimately bounded ultimate bound tracking error made arbitrarily small appropriate choice design parameters restricted control input saturation best authors knowledge first time robust attitude tracking controller designed aerobatic helicopter retains rotor dynamics globally defined classifies uncertainties involved two classes appendix proof lemma proof positively invariant sets since along solutions negative definite respective boundaries leads valid therefore continuously decreases enters sublevel set finite time leads hence ultimate bound references references pieter abbeel adam coates andrew autonomous helicopter aerobatics apprenticeship learning international journal robotics research bilal ahmed hemanshu pota flight control rotary wing uav using adaptive backstepping control automation icca ieee international conference pages ieee bilal ahmed hemanshu pota matt garratt flight control rotary wing uav using backstepping international journal robust nonlinear control francesco bullo andrew lewis geometric control mechanical systems modeling analysis design simple mechanical control systems volume springer science business media robert chen effects primary rotor parameters flapping dynamics emilio frazzoli munther dahleh eric feron trajectory tracking control design autonomous helicopters using backstepping algorithm american control conference proceedings volume pages ieee vladislav gavrilets emilio frazzoli bernard mettler michael piedmonte eric feron aggressive maneuvering small autonomous helicopters approach international journal robotics research marco beat gerig modeling guidance control aerobatic maneuvers autonomous helicopter phd thesis eth zurich hall bryson inclusion rotor dynamics controller design helicopters journal aircraft steven ingle roberto celi effects higher order dynamics helicopter flight control law design journal american helicopter society khalil nonlinear systems pearson education prentice hall isbn url https kufeld dwight balough jeffrey cross karen studebaker flight testing airloads aircraft annual forum helicopter society volume pages american helicopter society lee robust adaptive geometric tracking controls application attitude dynamics quadrotor uav arxiv august lee leok harris mcclamroch nonlinear robust tracking control quadrotor uav arxiv september robert mahony tarek hamel pflimlin nonlinear complementary filters special orthogonal group ieee transactions automatic control sanjeeva maithripala jordan berg wijesuriya dayawansa tracking simple mechanical systems general class lie groups ieee transactions automatic control marconi naldi robust nonlinear control miniature helicopter aerobatic maneuvers proceedings rotorcraft forum maasctricht netherlands pages lorenzo marconi roberto naldi robust full tracking control helicopter automatica bernard mettler identification modeling characteristics miniature rotorcraft springer science business media simone panza marco lovera rotor state feedback helicopter flight control robustness fault tolerance control applications cca ieee conference pages ieee nidhish raj ravi banavar abhishek mangal kothari attitude tracking control aerobatic helicopters geometric approach corr url http ioannis raptis kimon valavanis wilfrido moreno novel nonlinear backstepping controller design helicopters using rotation matrix ieee transactions control systems technology marc takahashi helicopter flight control law design without rotor state feedback journal guidance control dynamics shuai tang qiuhui yang shaoke qian zhiqiang zheng attitude control helicopter based backstepping proceedings institution mechanical engineers part journal aerospace engineering bing zhu wei huo robust nonlinear control helicopter parameter uncertainties nonlinear dynamics
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competition complexity auctions result bidders alon michal ophir inbal dec matthew december abstract seminal result bulow klemperer demonstrates power competition extracting revenue selling single item bidders whose values drawn regular distribution simple vcg mechanism case second one additional bidder extracts least much revenue expectation optimal mechanism beauty theorem stems fact vcg priorindependent mechanism seller possesses information distribution yet recruiting one additional bidder performs better mechanism tailored exactly distribution hand without additional bidder work establish first full results environments proving recruiting additional bidders revenue vcg mechanism surpasses optimal possibly randomized bayesian incentive compatible mechanism given environment bidders term number additional bidders needed achieve guarantee environment competition complexity using recent framework cai reasoning optimal revenue show competition complexity bidders additive valuations independent regular items least log extend results bidders additive valuations subject constraints showing significantly general valuations increase competition complexity additive factor improve bound special case matroid constraints provide additional extensions well computer science university alonarden work done part author visiting simons institute theory computing computer science university microsoft research computer science university ophirfriedler computer science hebrew university jerusalem inbaltalgam computer science princeton university smweinberg work done part author research fellow simons institute theory computing introduction great deal research recent years devoted design simple mechanisms motivated part desirability practical settings see many subsequent works different measures mechanism complexity considered including computation communication complexity well simplicity mechanism description dependence details economic environment last measure motivated early work come known wilson doctrine simple mechanisms preferred complex settings order alleviate risks introduced various assumptions indeed wilson considered progress game theory depend freeness assumptions idea reflected distinction mechanisms cases assumed exist populations prior distributions bidders drawn difference prior dependent mechanisms priors details assumed fully known contrast designer mechanism assumes knowledge priors terms practical applicability mechanisms risk overfitting particular beliefs whereas mechanisms inherently robust course could get exactly guarantees mechanisms ones case welfare maximization vcg mechanism would simply demand mechanisms provably case revenue optimization best mechanisms strictly outperform best ones case myerson mechanism game becomes understand tradeoff simplicity optimality one approach direction achieved remarkable success past decade apply lens approximation address questions form fraction optimal revenue achieved simple mechanism works certainly offer explanation ubiquity simple mechanisms practice quite tell whole story lens approximation views environment fixed tries find suitable auction generates much revenue possible settings particularly markets small constant fractions amount large losses approximation guarantees even matching impossibility results may unsatisfactory however enter realm questions reason view environment fixed seller could instance spend additional effort recruiting extra bidders attend auction instead treating number participants given indeed typical economist might reasonably demand optimal revenue settle nothing less instead pose question seeking minimum cost change environment simple mechanism achieves guarantee beautiful result spirit theorem asserts sale single item symmetric bidders whose valuations drawn regular distribution running simple vcg mechanism auction case bidders extracts least much revenue expectation myerson optimal mechanism bidders beauty theorem stems fact vcg simple standard mechanism completely oblivious bidder distribution yet recruiting single additional bidder revenue surpasses optimal auction original result made possible work myerson thanks works study prior independence others propose mechanisms inherently simple robust sell item revenue optimization extremely settings contrast revenue optimization extremely settings even additive bidders least recently owing complicating factors randomization computational intractability factors arise settings results competition complexity paper establish first full results wide range settings full results mean theorem statements form revenue mechanism symmetric bidders drawn population least large expectation optimal revenue bidders long satisfies condition simple paper restrict vcg original result revenue required surpass optimal revenue without additional bidders approximation loss whatsoever optimal revenue refers optimum among possibly randomized bayesian incentive compatible bic bayesian individually rational bir minimum full statement holds environment number bidders number items class valuations distributions say environment competition complexity competition complexity thus measures much competition needed revenue simple mechanism reaches strong revenue benchmark competition complexity complements sense measure sample complexity measures much data needed revenue mechanism approximates benchmark results techniques state results remark even priori clear number additional bidders suffices reach optimal revenue rather approximate fact without independence assumption bidders distributions already exist distributions additive valuations two items revenue vcg finite number bidders never surpasses revenue optimal mechanism one bidder result direct corollary consider following distributions valuation functions every item independently drawn distribution every set items maxt set system valuations called additive subject constraints independent regular say independent regular items example additive valuation function also consider extensions constraints randomly drawn independently item values instances replace constraints randomly drawn constraints note encompasses class valuations bidders typically also independence regularity conditions distribution bic means equilibrium bidders bid true value bir means agents expected utility equilibrium contrast work uses strictly lower benchmark namely optimum among deterministic dsic mechanisms section bidders values correlated shows infinite gap revenue optimal randomized mechanism optimal deterministic one even one bidder two items implies finite bound competition complexity since revenue vcg item bounded optimal revenue item concave function number bidders revenue submodularity theorem obtain following competition complexity results competition complexity bidders additive valuations independent regular items least log competition complexity bidders additive valuations subject constraints independent regular items greater competition complexity additive bidders independent regular items exist constraints matroid set systems competition complexity bidders additive valuations subject matroid constraints independent regular items greater competition complexity additive bidders independent regular items maximum number disjoint independent sets span include item items exist constraints tight competition complexity bidders additive valuations subject randomly drawn constraints independent regular items greater competition complexity additive bidders independent regular items direct implication theorem competition complexity bidders additive valuations subject constraints independent regular items least note matroids valuations additive valuations final bullet accommodates markets instance bidders equally likely additive independently values techniques high level approach breaks three steps find suitable upper bound optimal revenue many additive bidders using duality framework prove additive bidders vcg additional bidders surpasses benchmark using coupling arguments probabilistic tools prove reduction arbitrary downwardclosed set systems additive bidders cost additional bidders step one revenue upper bound make use duality framework developed upper bounds derive additive bidders provably suffice goals specifically single additive bidder exists distribution additive two independent regular items revenue vcg bidders exceed bound example appears section first step provide new bound still using framework suitable setting without getting much detail bound viewed taking similar approach quantile space opposed value whenever approach compares two random variables value instead compare postpone details section note expect approach useful future results mechanism design general step two covering extra bidders finding suitable bound need show revenue vcg additional bidders surpasses bound key tools fact regularity replaced next item property distributions quantile random variable drawn distribution cdf coupling arguments random variables representing revenue vcg additional bidders random variable step one wish highlight unique challenge targeting full result opposed approximation postpone details argument section whereas settings closed formula optimal revenue optimal revenue multidimensional settings expectation extremely bizarre random variable tractable bound optimal revenue therefore necessarily loose meaning one needs extra careful avoid sacrificing moreover known bounds including others cai framework still obviously simple analyze simplify analysis cost one upper bound maximums sums indicator random variables etc instance decomposition follows sequence operations like since want full result simplifications lossy necessarily complicating analysis surprisingly relevant example consider two random variables expectation imagine wish design auction gets revenue least max willing lose factor two simply design auction gets revenue least done really refuse lose constant factors forced design better step three reduction arbitrary feasibility constraints additive bidders believe result additive bidders already exciting right able extend far general settings reduction specifically first show revenue optimal mechanism additive bidders upper bounds optimal revenue bidders additive subject constraints assuming item values drawn distributions show revenue vcg bidders additive subject constraints least large revenue vcg additive bidders assuming item values drawn distributions chaining inequalities together conclude competition complexity increases relative additive bidders note similar result bidders naturally extended work additive bidders subject matroid constraints entirely new combinatorial approach necessary arbitrary constraints result may independent interest moreover reduction allows future research focus exclusively improved results additive bidders immediately extend far generally computational remarks well known running vcg general environments results hold see still wish note following even result additive bidders first full result beyond settings vcg implemented setting fact whenever feasibility constraints form matroid even environments implement vcg competition complexity meaningful measure right settings motivate full results versus approximations may exactly practitioners find way solve problems high stakes instances anything less incidentally encounter exact obstacle highest virtual value capture expected revenue auction obvious one design auction achieving revenue max see section optimum unsatisfactory bullets explain important understand competition complexity vcg even computationally intractable still provide additional competition complexity results appendix computationally efficient mir mechanism specifically show mir mechanism optimizes space matchings done also witnesses competition complexity bidders additive valuations subject constraints independent regular items related work results related work proves previous result result holds bidders independent regular items compares optimal deterministic dominant strategy incentive compatible dsic bounds optimal revenue known due due weaker benchmark able bound competition complexity tight variants results settings established rich literature mechanisms settings recently growing literature also considering mechanism design limited samples population whereas mechanisms get zero samples comparison present paper results either study settings incur approximation loss related techniques already discussed use recent duality framework develop revenue upper bounds concurrent work framework utilized prove approximation results simple mechanisms settings many xos bidders independent items single bidder values exhibit limited complementarity present authors dynamic auctions step one approach bears similarity comparable steps works work addresses unrelated problems techniques specific problem hand also worth noting merely provides framework deriving upper bounds attainable revenue upper bound used always indeed judging recent applications framework even selecting appropriate bound given setting seems bit art although approach across different works course similar related themes tcs results resemble spirit ideas beyond complexity particular concept resource augmentation idea compare performance algorithm endowed resources optimal outcome environment less resources examples results also referred bicriteria results appear online paging network routing games truthful job scheduling buffer management mechanism dsic ever bidder interest tell truth matter bidders report known bic mechanisms strictly outperform best randomized dsic mechanism settings organization preliminaries appear section section section address single additive bidder symmetric items whose values drawn single distribution simple case presents key idea guides develop new section using framework section present main results multiple additive bidders section generalizes results additive subject feasibility constraints additional generalizations presented sections matroid asymmetric feasibility constraints section lists open research directions preliminaries heterogeneous items bidders every item associated distribution support bidder value item vij drawn independently distribution write let density write let denote myerson virtual value work assume regular every monotone consider bidders additive subject feasibility constraints bidder feasibility constraint set system bidder value set given vij max vij every bidder say bidders symmetric respect feasibility constraints bidder valuation additive sometimes refer additive bidders unconstrained bidders let denote value profile agents sampled let vnj values agents item let denote values sampled agent without item given vector real values let denote highest value let max let denote indicator random variable event auction design mechanism pair allocation payment functions functions submitted bids use denote bider possibly randomized payment function standard study mechanisms bayesian individually rational bir mechanisms expected utility agent bayesian incentive compatible bic mechanisms maximizes agent expected utility cases expectation randomness mechanism agents valuations strategies given value distribution denote rev thep expected revenue optimal mechanism mechanism maximizes mechanisms single item sale second price auction lazy reserves first sets reserve prices bidder solicits bid bidder allocates item highest bidder bid surpasses case payment maximum second highest bid item remains unallocated continuous distributions distribution let denote competition complexity let opt denote revenue optimal mechanism bidders valuations drawn let also rev denote expected revenue mechanism played bidders valuations drawn use term competition complexity formally mean definition competition complexity respect mechanism bidders class distributions minimum opt rev vcg drop respect definition classes distributions considered additive independent items distribution additive independent items exist distributions fmp drawing amounts drawing setting subject constraints distribution additive subject constraints independent items exist one dimensional distributions drawing amounts drawing setting maxt subject randomly drawn constraints distribution additive subject randomly drawn constraints independent items exists distribution set systems distributions drawing amounts drawing independently setting maxt referenced regular say independent regular items classic theorem language competition complexity competition complexity bidders valuations single regular item discrete continuous distributions like explicitly consider distributions finite support like results results immediately extend continuous distributions well via discretization argument refer reader formal statement proof tie breaking throughout paper assume ties values agents ties break lexicographically first agent item tie breaking also assume exists unique welfare maximizing allocation items agents given two allocations welfare break ties symmetric difference according agent item duality benchmarks equation presents upper bound rev single bidder details reader referred appendix particular let event let event rev single bidder section give upper bound competition complexity single additive bidder results convey intuition regarding techniques used proving results complex settings benchmark use order prove results symmetric items consider single additive bidder pidentically distributed items therefore let myem expected revenue selling item separately bidders using myerson optimal mechanism write sum items even though identically distributed section describe proof changes items necessarily identically distributed lemma myem rev proof consider following mechanism selling item bidders run second price auction personal lazy reserve price bidder lazy reserve bidders let denote value profile bidders analyze expected revenue mechanism consider revenue bidder random variable denoted separately revenue bidders denoted optimality myerson mechanism holds myem couple simply setting consider first myerson theorem expected revenue bidder equals expected virtual surplus coupling since items identically distributed bidder allocated mechanism precisely probability event occurs bidder virtual surplus get consider next using fact second price auction bidders win whenever event occurs pay least comparing bounds summed items equation completes proof corollary competition complexity single bidder whose valuation additive regular items proof consider revenue vcg additional bidders additivity vcg separable items therefore revenue exactly sum second price auctions run item separately bidders applying classic theorem item separately get vcg bidders gives revenue greater revm rev last inequality follows lemma note proof extend items identically distributed reason probability bidder wins single parameter environment identical probability event happens fact next present example previous benchmark used fails provide meaningful bounds competition complexity items values drawn different distributions example previous benchmark fails proposition every exists value distribution additive independent regular items rev vcg less benchmark cai one bidder valuations drawn equation proof consider market single bidder two items item distributed according equal revenue distribution capped value supported item distributed according following cdf supported one verify distributions indeed regular consider benchmark proposed case two items interpreted expected virtual value item highest value two plus expected value item lower value two since item always higher value item exactly expected virtual value item plus expected value item expected value item expected virtual value item one readily compute combining expected value item expected virtual value item get benchmark least consider running vcg using bidders total additional bidders since running vcg additive bidders running vcg item separately analyze much revenue get running vcg item first analyze revenue running myerson optimal mechanism item bidders upper bound running vcg expected revenue optimal auction revenue submodularity expected revenue running optimal mechanism concave function number bidders thus expected revenue optimal auction bidders compute revenue optimal mechanism item bidders single bidder optimal mechanism sets price item gets expected revenue two bidders optimal mechanism vcg monopoly reserves since monopoly reserve equivalent running vcg compute expected revenue vcg following way fix bidder compute expected payment bidder value sets random threshold bidder quantile bidder value bidder expected payment exactly probability value times exactly therefore expected payment bidder exactly area revenue curve equal symmetry bidder expected payment well expected revenue overall submodularity revenue since second bidder adds expected revenue expected revenue vcg bidders get expected revenue obtained running vcg items bidders therefore least order cover one easy way shortcut computation apply myerson theorem revenue virtual welfare immediately conclude figure revenue curve distribution used example benchmark since arbitrary large benchmark suitable providing meaningful bound competition complexity market asymmetric items section present new upper bound optimal revenue demonstrate applicability showing implies competition complexity result case asymmetric items additive bidder new bound quantile based regions previously discussed proof case items sampled distribution extend items reason probability jth sample highest one probability bidder highest value item since samples bidders values item values single bidder different items original scenario intuition behind new benchmark use want mimic bidders samples even items item define item region bidder region defined region bidder value highest quantile according items distributions using definition give new upper bound revenue theorem new upper bound rev every product distribution additive bidders whose valuations sampled rev max vij ivi vij ivi let event let event direct corollary theorem gives following revenue single bidder corollary every product distribution revenue single additive bidder whose valuation sampled upper bounded one may notice probability event equals exactly probability samples jth sample highest derivation new upper bound uses framework resulting corollary deferred appendix particular corollary stems application single additive bidder asymmetric items new bound hand show sidestep impossibility result proposition associated upper bound theorem competition complexity single bidder whose valuation additive independent regular items proof proof almost identical proof section state changes proof proposition use upper bound vector draws whereas vector independent draws couple every coordinate still bidder allocated mechanism precisely probability event occurs virtual value positive using monotonicity iff second price auction bidders win whenever event occurs pay least main result multiple additive bidders section given instance additive bidders heterogeneous item bidders valuations drawn product regular distributions show adding bidders running vcg able get least much revenue optimal obtainable revenue bic mechanism original bidders theorem competition complexity bidders additive valuations independent regular items least log remainder section contains proof upper bound missing proofs claims deferred appendix lower bound proved appendix makes use equal revenue example upper bound item contribution revenue fix item consider contribution upper bound revenue given revj max vij ivi vij ivi given valuation profile let index bidder ith highest valuation item next key lemma gives bound revj useful proving result lemma revj upper bounded prn max prn proof consider two cases vij vij ivi vij ivi every bidder therefore max vij ivi vij ivi hand max vij ivi vij ivi max max vij ivi vij ivi max max get revj max max single parameter lemma section prove main technical lemma lemma consider case single item sale bidders whose values drawn expected revenue obtained running vcg item least revj first show proving lemma immediately yields proof theorem proof theorem based lemma additive bidders running vcg equivalent running vcg item separately applying lemma get rev revj vcg vcg proving lemma make use following property vcg folklore knowledge proof appears appendix completeness observation folklore consider set bidders drawn regular distribution single item sale optimal mechanism always sells item vcg mechanism order prove lemma introduce mechanism selling single good set bidders drawn show expected revenue mechanism least bound revj moreover mechanism always sells item therefore observation vcg gets least much revenue bound revenue mechanism sufficient prove lemma mechanism given fig input bids sampled distribution consider bidders arbitrary predetermined order rename bidders follows rename first bids ith highest bid first bids let denote vector rename next bids according arbitrary order let denote vector rename last bids ith highest bid last bids let denote vector let arg allocate item else allocate item else allocate item charge winner according myerson payment identity figure single parameter mechanism expected revenue least revj first note mechanism truthful since allocation every bidder monotone bidder valuation giving full proof lemma give intuition trick trying couple various events etc affect random variable related revj events determine different cases mechanism reader get good idea events coupled based decision allocate item however complete analysis bit subtle particular requires following probabilistic claim proof appears appendix addition proper coupling claim regular distribution max max note lhs random variables correlated come bidders rhs independent comes bidders proceed prove main technical lemma proof lemma consider values upper bound revj defined renaming phase couple values whenever every see valid coupling observe vectors samples sampled uniformly interval independent samples next analyze expected revenue obtained bidders compute expected virtual value bidders compute expected payment distinguish two cases case condition checked step mechanism case item goes bidder since bidder pays minimum value allocated payment least probability case second equality follows coupling last equality follows definition independence samples case event happens complementary probability case case winner determined according condition step mechanism winner bidder otherwise winner payment identity payment least combining two cases expected revenue least bound summand bound separately first summand second equality follows coupling last equality follows definition independence samples second summand first notice max max max max max inequality follows claim first equality follows coupling last equality follows definition independence samples consequently revenue least max greater revj according lemma since revj mechanism always sells item applying observation completes proof lemma generalizations additive subject section generalize results beyond additive bidders bidders whose valuations additive subject constraints section consider distributions constraints fixed entire distribution section state following extensions proofs appendix special case constraints matroids obtain improved guarantees competition complexity theorem proved appendix case constraints part distribution additive subject randomly drawn constraints obtain slightly weaker guarantees theorem proved appendix finally extend results competition complexity respect mechanism complements results respect vcg settings vcg implemented theorem proved appendix symmetric general constraints section bidders valuations additive subject identical feasibility constraints represented set system assume without loss generality every item main result section following theorem competition complexity let competition complexity additive bidders independent regular items competition complexity additive bidders subject constraints independent regular items theorem relies lemma corresponding lemmas matroids asymmetric feasibility constraints appear appendices respectively particular lemmas state main lemma use following notation fix product regular distributions let vcgadd denote expected vcg revenue selling items adn ditive bidders whose values draws let vcgdc denote expected vcg revenue selling bidders values drawn whose valuations additive subject identical feasibility constraints intuitively expected revenue vcg unconstrained bidders higher revenue vcg constrained bidders since former bidders compete items lemma gives bound much larger relative intuition ceases hold add lemma main lemma vcgdc vcgn proof lemma appears section lemma thought quantifying extra competition complexity required due feasibility constraints bidders comparison unconstrained additive bidders main technical hurdle proving theorem high level proof follows first observe however vcg decides allocate items always option reallocate item bidder currently receives nothing whoever receives item pays least highest value item among bidders receive nothing trick comparing random variable bidders value bidders due potential complexity arbitrary feasibility constraints random variable denoting highest value item among bidders receive nothing depends quite intricately values bidders items one reason random variable arguments one might use feasibility constraints matroids careful combinatorial argument required instead given lemma proved shortly prove main theorem section proof theorem fix product regular distributions feasibildenote optimal expected revenue ity set system use following notation let revadd achieved selling items additive bidders whose values draws denote optimal expected revenue selling bidders whose let revdc valuations subject constraint note first every feasibility set system revadd revdc follows following three facts without loss generality optimal mechanism agents subject constraints allocates sets designer mechanism additive bidders free restrict allocating sets subject restriction immaterial whether bidder additive additive subject particular exists optimal mechanism bidders constrained truthful unconstrained bidders optimal mechanism get better assumption competition complexity additive bidders add add know vcgadd revn apply lemma get putting everything together add add vcgdc revdc revn shown competition complexity additive bidders subject constraints completes proof theorem proof main lemma first introduce lemma gives lower bound revenue vcg bidders subject identical constraints lemma consider set bidders identical feasibility constraint outcome vcg item let highest value unallocated bidder item bidder receive item revenue vcg least proof recall vcg mechanism bidder pays externality environment difference welfare bidders allocated let respective sets allocated unallocated bidders fix bidder let feasible set items receives vcg let welfare vcg outcome welfare bidders allocation consider following allocation allocate items allocation vcg mechanism distribute items among unallocated bidders item allocated bidder value item notice since bidders allocated items receive subsets feasible set items allocation also feasible due feasibility set welfare purposed allocation since described allocation items bidders welfare optimal allocation bidders without least high get payment bidder least summing bidders completes proof lemma add goal show vcgdc vcgn recall additive bidders vcg add decomposes items therefore vcgn sum expected revenues obtained selling every item separately bidders whose values drawn using vcg auction case vcg equivalent auction prove lemma thus sufficient show every item expected revenue selling separately bidders expectation second highest among random samples denoted expected payment item vcg constrained bidders contribution selling item vcgdc remainder proof shall argue vcg constrained bidders always least bidders whose values item draws single bidder set allocated since lemma payment item least highest value among unallocated bidders show expected payment completing proof fix item fix values constrained bidders items consider allocation items bidders denote allocation set allocated bidders clearly since allocated items notice following claim item allocated bidder bidder allocated item proof argue allocated bidder allocation items opt identical allocation precisely one bidder allocated vcg required items allocated due constraints bidders additive valuations show assuming otherwise leads contradiction consider allocation assume contradiction bidder allocated item get allocation opt need perform additional reallocations items since assume unique optimal allocation described section means need additional reallocations get optimal welfare valuations additive subject constraints increase total welfare item reallocations bidder item get increase welfare reallocations bidder item contradicts optimality completes proof claim fix values bidders item remaining randomness values bidders item information currently allows compute optimal allocation optimal allocation allocated bidder denote allocation importantly lets identify set bidders size least allocated allocated bidder claim sampling values item set one identify set none bidders allocated case item allocated bidder opt proof whenever allocated bidder mentioned computed even though know value item bidders simply compute allocation allocated bidder prove existence unallocated bidders proof following charging argument show chain reallocations starts allocation leads allocation argue ends least bidders unallocated start bidder gets item say bidder vacates item allocated say bidder snatches item allocated adds allocation let vacate snatch items allocation reaches allocation continue chain place following bidders queue provided already queue snatched item end grabbing item vacated importantly bidder enters queue unique item entrance charged either item snatched item vacated grabbed take next bidder queue repeat process letting bidder vacate snatch items reaches allocation placing bidders queue described queue empty notice assumption unique optimal allocation bidder entered queue allocation true since items bidder snatched bidders otherwise would contradict optimality use chain reallocations show mapping bidders items allocated bidders mapping follows bidder entered queue since item snatched different bidder mapped snatched item bidder entered queue since grabbed vacated item mapped item grabbed bidder entered queue mapped item allocated mapping implies bidders least items allocated item allocated bidder bidders mapped distinct item allocated bidder implies items allocated since must hold bidders unallocated using two claims prove lemma proof lemma let set bidders guaranteed exist claim take arbitrary bidder add bidder guaranteed exist since mentioned set least bidders whose value item yet sampled values bidders item draws required known principle deferred decision complete proof claim one bidders allocated opt consider two cases allocated bidder claim bidder allocated item opt therefore claim follows allocated bidder claim bidders different unallocated opt claim follows implies payment item vcgdc least second highest draws exactly payment item vcgadd completes proof lemma therefore theorem extensions special case constraints matroid obtain improved bounds competition complexity proof definition disjoint spanning number appear appendix note disjoint spanning number always theorem competition complexity matroids let competition complexity bidders valuations additive independent regular items let matroid disjoint spanning number competition complexity additive bidders valuations additive subject constraints independent regular items distributions additive subject randomly drawn constraints obtain following slightly weaker guarantee competition complexity proof appendix theorem competition complexity randomly drawn downward closed constraints let competition complexity bidders additive valuations independent regular items competition complexity bidders additive valuations subject randomly drawn constraints independent regular items finally extend results competition complexity respect mechanism specifically consider mechanism considers allocations matchings allocate bidder one item denote mechanism vcgud note vcgud implemented proof appears appendix theorem competition complexity respect mir mechanism let competition complexity additive regular bidders competition complexity respect vcgud bidders additive valuations subject randomly drawn constraints independent regular items discussion future work present first full results settings show competition complexity buyers additive valuations subject feasibility constraints independent regular items least special case additive constraints competition complexity least log traditional sense easy corollary fixed vcg guarantees optimal obvious open question close gaps bounds least special case additive even concretely right dependence additive bidders linear logarithmic dependence necessary latter especially enticing answer vcg approaches optimal revenue another direction consider mechanisms order get better guarantees instance maybe log lower bound circumvented considering different mechanism perhaps better two prior independent mechanisms overall believe results already quite substantial strongly feel wealth open problems direction competition complexity references azar daskalakis micali weinberg optimal efficient parametric auctions proceedings annual symposium discrete algorithms society industrial applied mathematics azar kleinberg weinberg prophet inequalities limited 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optimal auction design mathematics operations research pavlov property solutions linear monopoly problems journal theoretical economics rochet chone ironing sweeping multidimensional screening econometrica roughgarden schrijvers ironing dark proceedings acm conference economics computation maastricht netherlands july roughgarden yan robust auctions revenue via enhanced competition working paper preliminary version appeared roughgarden tardos bad selfish routing journal acm jacm rubinstein weinberg simple mechanisms subadditive buyer applications revenue monotonicity proceedings sixteenth acm conference economics computation acm segal optimal pricing mechanisms unknown demand american economic review sleator tarjan amortized efficiency list update paging rules commun acm thanassoulis haggling substitutes journal economic theory vickrey counterspeculation auctions competitive sealed tenders finance welsh matroid theory courier corporation wilson analyses trading processes advances economic theory fifth world congress yao solutions maximum revenue auction dominantstrategy bayesian implementations yao bidder reduction auctions applications proceedings annual symposium discrete algorithms siam duality framework reduced forms reduced forms defined probabilities type spaces type space agent reduced form auction stores bidders items types probability agent receive item reporting mechanism randomness mechanism randomness agents reported types assuming come well expected price bidder pays reporting type randomness easy see buyer additive subject constraints expected value reporting type mechanism long mechanism ever awards type bidder set items fit constraints whenever type bidder subject constraints mechanism awards subset items always exists optimal mechanism property furthermore trivial transformation turn mechanism without property one property achieves exactly revenue see say reduced form feasible exists feasible mechanism matches probabilities promised reduced definition reworded mapping definitions given corresponding virtual value function transformation vectors valuation vectors closure linear combinations satisfies equipped definitions state main theorem framework theorem revenue virtual welfare let flow conserving bic bir mechanism revenue virtual welfare virtual value function corresponding therefore task finding good bound reduces finding good flow conserving duals order find flow conserving duals begin following definition definition region region subset types upward closed respect item every types vij vij vij also use denote region respect item method designing flow conserving duals following first partition type space regions define duals follows type region upward closed let vij type vij largest value smaller vij exists also set otherwise set set duals let resulting virtual transformation following lemma essentially restatement claims upwards closed regions comprehensive description reduced forms applications see equivalent stating exists might look similar wording definition lemma let valuation vik vij otherwise using flow assuming item exactly one region associated case throughout entire paper get following upper bound revenue mechanism theorem set regions optimal revenue seller obtain bic bir mechanism rev max max vij ivi vij ivi vij ivi vij ivi proof inequality direct corollary theorem lemma equality follows since max achieved whenever vij single additive bidder case since single additive bidder feasibility constraints bound simplifies rev regions defined follows contains valuations every breaking ties lexicographically therefore get following bound rev instead define regions follows contains valuations every breaking ties lexicographically therefore single bidder bound form rev multiple additive bidder case present proof theorem multiple additive bidder benchmark use prove main result proof theorem notice equation since feasibility constraints allocate item maximizing virtual value therefore rev max vij ivi vij ivi max vij ivi vij ivi max vij ivi vij ivi missing proofs section proof observation since expected revenue mechanism expected virtual value allocated bidder optimal mechanism always sells item allocates item bidder highest virtual value since bidders drawn regular distribution bidder highest virtual value also bidder highest value thus optimal mechanism always sells item allocates item bidder highest value exactly allocation rule vcg proof claim proving claim introduce following definitions lemmas definition random variable stochastically dominates fosd random variable every definition consider two random variables say positively correlated every fosd every lemma positively correlated independent fosd max max proof prove claim exactly marginals buth independent since fosd independent immediate max max max desired first notice max max similarly max second equality holds since marginals subtracting yields max max third equality follows fact marginals independence inequality follows fact positively correlated lemma regular distribution sampled respectively positively correlated proof fix value let highest value virtual valuation function first last equalities follow regularity last inequality follows fact whenever also immediately implies positive correlation lemma every distribution sampled sampled fosd proof fix value support need show given vector sampled let first coordinates last coordinates notice event resides first coordinates independent event therefore inequality follows since whenever also last equality follows renaming combining aforementioned three lemmas three lemmas yields claim proof claim consider random variables lemma positively correlated lemma fosd addition independent claim follows lemma additive subject matroid feasibility constraints section establish improved version lemma special case matroids use following notation given product regular distributions bidders subject identical matroid feasibility constraints let vcgmat denote expected vcg revenue selling items bidders values drawn whose valuations additive subject matroid constraint let matroid set system rank function set spans element rank increase added introduce following parameter matroid call matroid disjoint spanning number definition disjoint spanning number element matroid let disjoint spanning collection family disjoint independent sets every spans let denote maximum size disjoint spanning collections define maxj example state value two kinds simple matroids example examples disjoint spanning numbers matroids partition matroids size largest partition state main lemma matroids shows extra competition complexity arising imposing matroid constraints additive bidders lemma main lemma matroids every bidders matroid feasibility add constraint whose maximum disjoint spanning number vcgmat vcgn proof theorem proof theorem lemma replaced lemma proving lemma briefly discuss relation work special case bidders valuations prove version lemma case bidders bidders alternatively described additive bidders subject matroid constraints particular add show vcgud precisely get apply lemma matroids using disjoint spanning number matroids equal example fact inequality shown lemma general example graphical matroid would maximum number cycles share edge otherwise disjoint constraints conclude case bidders hardest one terms competition complexity among matroids also among set systems generalize result two ways first extending constraints lemma second introducing parameterized version matroids using environment disjoint spanning number parameter lemma terms techniques result matroids uses stability argument similar result general uses different charging argument described section results use principle deferred decision used analysis order argue values unallocated bidders determine vcg payments sufficiently high expectation proof main lemma matroids next claim used proof lemma follows easily definitions matroid rank function spanning matroids claim consider matroid every subset elements every two elements spans span span moreover every subset span state following claim gross substitutes valuations apply proof lemma subclass additive valuations subject matroid constraints claim uses language vacated snatched items defined proof lemma claim consider allocation items bidders gross substitutes valuations allocation opt items bidders found chain reallocations starting step bidder snatches either item first step single vacated item subsequent steps possibly vacates another single item proof sketch let vector walrasian prices supporting allocation bidder allocated demand bundle items prices unallocated item priced add item initial price prohibitively high show process transforms opt process establishes existence chain reallocations claimed computational properties irrelevant context start lowering bidder longer getting demand reaches found opt see observe since bidder allocated demand given current prices item positive price unallocated reached walrasian equilibrium whose allocation first welfare theorem otherwise single improvement property gross substitutes since prices support allocation items bidder improve utility snatching single item whose price reduced first step item possibly vacating another single item item vacated reached equilibrium allocation opt otherwise item vacated repeat process lowering price item process must end allocation opt thus completing proof sketch assuming ties simplify exposition full formality lowered taken prove main lemma matroids add proof lemma goal show vcgmat vcgn proof lemma sufficient identify constrained bidders whose values item draws vcg payment least high second highest value bidders establishes contribution selling item vcgmat least exactly contribution selling item vcgadd proof lemma fix item fix values constrained bidders items denote allocation items constrained bidders diverging proof lemma denote set bidders whose allocation spans notice fix values bidders item remaining randomness values bidders item random samples denote opt random allocation items including bidders assume first item allocated opt bidder proof lemma allocation items change opt since bidders still allocations span item vcg payment least high value bidders assume item allocated opt bidder note proof lemma find opt without knowing several allocations assume vcg finds one implied claim claim know bidders changed allocation single improvements involving snatching item possibly vacating another fact single improvement one item vacated since bidders valuations additive subject matroid constraints bidder allocation vacates item first snatches item spans claim means eventual allocation bidder opt still span item conclude vcg payment least high value bidders get cases payment least value bidders since set determined sampling values item completes proof lemma additive subject asymmetric downward closed feasibility constraints section consider case heterogeneous feasibility set systems bidders let denote feasibility set system bidder results hold add mild assumption every bidder item first point part proof symmetric feasibility constraints fails unlike case bidders symmetric feasibility constraints payment bidder lower bounded sum highest values unallocated bidders items contrast lemma therefore need new lower bound revenue vcg stating new lower bound introduce following notation order items lexicographic order given set define see consider additive bidder allocated two items vcg mechanism two unit demand bidders payment additive bidder arg bidder set highest value item contains bidders except arg value item bidder give lower bound revenue vcg given definitions lemma consider set bidders asymmetric feasibility constraint outcome vcg let set unallocated bidders assuming number bidders least revenue vcg least proof let respective sets allocated unallocated bidders let allocation returned vcg let welfare allocation fix agent received item welfare agents outcome consider following allocation allocate items distribute item unallocated bidder notice well defined since number agents assumed least construction every therefore bidder gets single item feasible assumption every welfare proposed allocation since described allocation items bidders welfare optimal allocation bidders without least high vcg agent pays externality agents loss welfare agents due existence get payment bidder least summing bidders completes proof lemma similarly proof theorem corollary prove lemma shows many extra bidders needed revenue vcg arbitrary downward closed constrained bidders exceed revenue vcg unconstrained bidders end fix product expected vcg revenue distribution regular distributions denote vcgasym selling items bidders values drawn whose valuations additive subject identical feasibility constraints lemma every bidders arbitrary downward closed feasibility constraint add vcgasym vcgn lemma useful right also used imply competition complexity result extending theorem case feasibility constraints fixed entire population prove lemma begin showing useful property notations used order bound vcg revenue lemma two sets bidders every proof proof suppose claim true consider two cases denote bidder induction hypothesis otherwise induction hypothesis thus therefore corollary get corollary two sets bidders every proof lemma therefore definition holds prove main lemma bidders asymmetric feasibility constrains proof lemma let set bidders unallocated vcg constrained bidders prove every expected value least high expectation second highest value draws argument suffices prove lemma since latter exactly paid item vcg unconstrained bidders previous proofs proof uses principle deferred decisions consider first sampling values items sample value item bidders allocated item optimal allocation allocate similarly proof lemma identify set bidders single bidder going allocated vcg furthermore value item agents yet determined denote random set left unallocated putting item back consider set bidders claim matter bidder allocated final allocation bidders remaining otherwise would bidder contradicting lemma notice set size therefore set size least furthermore set determined sampling values item bidders since single bidder one two bidders highest value item therefore corollary least payment item vcg unconstrained bidders applying lemma gives revenue vcg constrained bidders least revenue vcg constrained ones desired proof theorem first draw constraints independently bidder constraints drawn independently values apply lemma conclude constraints could possibly drawn expected revenue vcg bidders subject asymmetric constraints least expected revenue additive bidders conditioned drawn constraints therefore taking expectation randomly drawn constraints theorem statement holds competition complexity maximum range vcg furthermore show overcome intractability vcg general feasibility constraints even constraints asymmetric across agents recall vcgud defined run vcg restrict outcomes matchings every agent gets single item mechanism hence truthful furthermore since finding maximum weight matchings tractable also prior independent mechanism ask following question competition complexity mechanism well defined first provide lower bound revenue vcgud observation consider set agents asymmetric feasibility constraint outcome vcgud item let highest value unallocated agent item revenue vcgud least proof similarly proof lemma follows taking optimal allocation removing agent allocated item allocating item unallocated agent highest value item notice use observation essentially identical proof proof lemmas similar claims used show following lemma every bidders asymmetric downward closed feasibility constraints vcgud vcgadd turn yields following upper bound competition complexity vcgud number extra bidders one needs add original environment order get revenue greater optimal revenue original environment using mechanism vcgud complete proof theorem lower bounds competition complexity theorem competition complexity single bidder additive valuations independent regular items log proof consider single additive bidder items distributed according equalrevenue distribution cdf lemma shown optimal revenue setting satisfies rev log consider setting single item bidders distributed according expected revenue optimal auction revenue submodularity expected revenue running optimal mechanism concave function number bidders thus expected revenue optimal auction bidders actually shown exactly consequently expected revenue running vcg single item bidder setting upper bounded consider original setting single additive bidder items distributed according distribution based analysis linearity vcg additive bidders expected revenue running vcg additional bidders bidders thus order surpass optimal revenue rev number additional bidders least large log log establishes assertion theorem approximation vcg goes show direct corollary results running vcg large markets yields optimal revenue additive bidders theorem distribution regular items whenever running vcg bidders without recruiting additional bidders yields revenue half optimal revenue proof consider market additive bidders values sampled according theadd orems vcgadd revn since vcg decomposes across items additive bidders since revenue submodularity applies therefore vcgadd vcgadd add add revn revn
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city versus wetland predicting urban growth vecht area cellular automaton model baydinb eleveldc gilbertc apr faculty earth life sciences vrije universiteit amsterdam boelelaan amsterdam netherlands department applied physics chalmers university technology sweden institute environmental studies vrije universiteit amsterdam boelelaan amsterdam netherlands abstract many studies dealing protection restoration wetlands sustainable economic growth cities separate subjects study investigates conflict two area city growth threatening protected wetland area develop stochastic cellular automaton model urban growth apply vecht area surrounding city hilversum netherlands using topographic maps covering past years investigate dependence urban growth pattern values associated protected wetland types landscape surrounding city conflict city growth wetland protection projected occur assuming full protection wetland results also show milder protection policy allowing wetland sacrificed could beneficial maintaining valuable landscapes insight would difficult achieve analytical means conclude even slight changes usage priorities landscapes significantly affect landscape distribution near future results also point importance protection policy take value surrounding landscapes dynamic nature urban areas account keywords urban growth stochastic modeling cellular automata wetland landscape protection msc introduction landscape defined european landscape convention council europe zone area perceived local people visitors corresponding author email address atilim baydin preprint submitted elsevier april whose visual features character result action natural cultural human factors human activity one actors formation landscapes also cause destruction today unfortunately many landscapes throughout world serious threat due factors intense use land pollution insufficient regional planning wetlands marshy swampy lands type landscape direct threat current pattern human development even though wetlands historically considered type wasteland importance integral parts ecosystems become increasingly realized last decades whigham known strongly support wildlife habitats improve water quality buffer storms control erosion flooding kusler kentula besides high natural significance also high economic value natural resource areas recreation developed countries currently many environmental management policies effect protect wetlands trying permit utilization natural resource continue study intend analyze situation developing urban area lies close wetland protection urban growth practically inevitable sustainable economy henderson city resist growing protect wetland city threaten wetland wetland threaten city long feasible protect wetland investigate discuss questions case study vecht area netherlands home internationally renowned wetland city hilversum lying border protected area fig main aim investigate conflicting dynamics continuing urban expansion protection local wetland present urban growth projections based current distribution landscapes associated importance terms natural economic value using stochastic cellular automaton model model produces quantitative visual information changing distribution landscapes vecht area near future year used predict anticipated date clash growing city wetland also study outcomes several different scenarios different protection priorities believe insights similar models significantly valuable short long term regional planning works provide brief description area case study section followed details method employed computational model data sources section results three different runs model section conclusions drawn section study area vecht area urban development policy vecht area floodplain river vecht forming important part delta river system netherlands covers roughly rectangular figure overview vecht area rectangle nearby city hilversum major cities randstad conurbation green heart netherlands shaded area middle image landsat source usgs earthexplorer area includes wetlands small lakes streams patches create characteristic flora fauna high biological diversity history area last centuries unique combination natural economic processes early dominated peat extraction forming small ponds transformation larger lakes via erosion later recovery lost land via drainage regulation water tables area still dynamic structure terms economy nature trapped middle randstad conurbation considered important part national ecological network designated netherlands environmental assessment agency randstad also known deltametropool conurbation western netherlands increased agglomeration large cities underway formed mainly cities amsterdam leiden hague rotterdam utrecht hilversum cities surround green heart netherlands major open space vecht area part van eck idg fig gives overview location vecht area relative green heart randstad green heart provides valuable natural environment attractive leisure area population adjacent cities create new jobs ask growing urban development entire vecht area also part wet axis national ecological network outlined dutch nature policy plan lnv aims connect fragmented natural habitats improve region ecology work turn forms part larger plan natura european union protection seriously threatened habitats establishment special protection areas throughout europe lnv simultaneous high demand urban development nature preservation make task spatial planning urban areas region highly challenging although planning decisions applied strictly keep environmental quality high development sustainable progress yet insufficient noted yearly vinex reports ministry public housing spatial planning environment vrom vinex states lack communication cooperation municipalities dealing construction works provinces dealing connection natural habitats city hilversum since hilversum located close vecht area fig immediately surrounded valuable landscapes comprising natural wetlands wetlands arranged recreation areas also cornebos forest old forest attracting rare birds heathland westerheide vvv areas highly important terms ecological value arable lands pastures nearby also possess economic ecological importance pasturelands extent form transition zone wetland urban areas provide nesting areas many bird species complex structure surrounding landscape city planning hilversum strict city managed maintain stable population last years save protected landscape harm despite low population growth extent city keeps growing number new housings region suggested alternative measure city growth glaeser marshall fig shows data published statistics netherlands het centraal bureau voor statistiek cbs indicating clear trend increase number residential buildings even though population remains less steady period suggests phenomenon commonly referred urban sprawl effect area confirming kasanko method develop stochastic cellular automaton model urban growth built upon concept total importance value landscape onto city may grow model kept simple possible stage examine growth dynamics clarity implement model programming language provide core source code appendix model described section simplistic metric summarizing economic natural values associated piece land figure vecht area immediate neighbor hilversum landsat pseudocolor bands source usgs earthexplorer figure change population number residential buildings city hilversum last decade source statistics netherlands cbs statline available online http input assumptions making projections model first need adopt reasonable average annual rate urban growth assume investigating historical maps vecht area one get fairly accurate projection middleterm growth rate provided maps sufficiently detailed purpose analyze collection topographic maps region covering period majority belonging century table reason including recent maps selected period restrictions place protect natural habitats measured growth rate would reflect unrestrained factors affecting growth favorable would reproduce unbounded growth pattern city limiting pressure landscape protection dynamically imposed model simulation step top basis growth rate measurements define area city hilversum historical map supervised classification using software conclude annual urban growth rate around reasonable value assume observing trends growth historical data giving table list employed spatial data data source scale resolution date topographic map amsterdam hilversum dienst der militaire verkenningena topographic map topografische inrichting combined topographic map utrecht amersfoort netherlands ministry war topographic service satellite image geocovertm landsat nasab vecht area restoration plan van den bergh gilbert provided geudeke zandvliet accessible online https compressed mrsidtm color imagery weight trend latter half century value also checked projected economic growth rate statistics netherlands consistent defining boundaries protected wetland area model employ suggested restoration plan provided van den bergh gilbert use analyze evaluate scenarios wetland areas netherlands taking present situation account model simple discrete systems recurrently applied rules cellular automaton models demonstrated ability replicate important features complexity observed natural processes well number advantages continuous mathematical models well discussed literature schiff using models describing urban growth receiving increasing attention field urban planning study effects growth make predictions diverse geographic settings itami couclelis torrens sullivan model kind sleuth slope land cover exclusion urbanization transportation hillshade developed united states geological survey usgs incorporating factors effect transportation networks slopes addition spreading existing urban area jantz believe simpler model sufficient purposes study vecht area relatively small compared scales urban models frequently applied means effect excluding advanced features model insignificant another important factor characteristic flatness topography netherlands country including vecht area virtually void hills slopes rdg renders factors like slope resistance important part general urban models irrelevant case model consider area study grid cells cell one types urban wetland heathland forest pasture water cell types except water type urbanized initializing model current distribution landscapes region employ recent satellite image geocovertm set distributed nasa based landsat table process supervised classification technique cell types assigned real numbers ranging representing importance value means protection cells type value means total protection cells type circumstances year according annual growth rate number new cells appended existing urban area converting neighboring cells wetland heathland forest pasture urban cells done follows random cell selected grid checked whether least one urban neighbor random selection continues cell found selected cell converted urban cell probability inversely proportional landscape importance value assigned type every year simulation random selection conversion cycle continues required number new urban cells year created model permits formation diverse development scenarios assigning different importance values different landscape types flowchart fig gives outline model working number new urban cells created year simply number new urban cells annual growth rate number present urban cells annual growth rate urban area http type growth urban models new isolated urban cell occasionally formed without adjacent urban cells held fixed run based assumption expected growth rate city independent size cordoba figure flowchart employed urban growth model take cautious approach presuming ultimate date reasonable prediction model year thinking calculations covering longer periods would become diverted accumulation errors introduced assumptions selected parameters growth rate intrinsic limitations modeling approach yeh also trends priorities vecht area netherlands general likely changed beyond terms longer results discussion going back questions beginning city resist growing protect wetland part answered solely aid model literature experiences may help economic growth always desired direct influence growth cities similar situation described karen merced county united states area deal urban growth wetland recreation simultaneously california central valley exemplifies complex situation could become noted million per year spent maintenance wetland recreation area wetland contributes million per year back local economy provides jobs people paradox posed situation immediately seen effort protect wetland turns back contribution local economy turn promotes urban growth threatens wetland time wetlands recognized globally cornerstone focal point economic development developed developing countries ways sought balance protection use wetland restoration vecht area presents many valuable economic benefits nearby cities income increased tourism fishing city hilversum economic growth foreseeable growth planned way ensures survival wetland landscapes region city threaten wetland wetland threaten city answer question design first run model using parameters thought represent current policies priorities region wetland importance value set impossible destroy wetland make sure protected circumstances used parameter values summarized table main result simulation addition projected spatial distribution map amount loss area landscape type present resulting distribution different landscapes summarized table years amount area lost ratio projected area current area projected distribution landscapes presented fig years interval note annual urban growth rate runs assumed explained section predict clash wetland area city boundary become significant year well within timescale grant considering current recreation urban plans vecht area long wetland going protected results first run previous section indicate deficiency scenario wetland successfully protected comes cost losing valuable landscapes noticeable decrease figure first run model definite protection wetland decreasing importance values heathland forest pasture details table resulting distribution landscapes shown year intervals table parameters results first run definite protection wetland decreasing importance heathland forest pasturea landscape importance amount remaining lost original amount remaining lost original wetland heathland forest pasture annual growth rate simulation starts year total forest area almost original amount table one may argue proximity forest urban area makes prone depletion still possible prevent amount loss trying adjust protection priorities two runs model section designed investigate whether permitting loss wetland helps save greater amount forest purpose model run many times giving different importance values type landscape observing results present results two runs table fig present situation wetland importance parameter reduced effort prevent drastic loss forest first run surprisingly noticed little sacrifice wetland one potential save significant amount forest around remaining contrast figure first run compare table table results radical change protection policy presented table fig assign wetland heathland forest equal protection priorities results situation even distribution figure second run model wetland importance reduced effort prevent drastic loss forest first run details table resulting distribution landscapes shown year intervals table parameters results second run second run model landscape importance amount remaining lost original amount remaining lost original wetland heathland forest pasture annual growth rate simulation starts year resources achieved still maintaining wetland others issue recognize achieved wetland assigned importance value landscapes result particular geographical distribution landscapes region forest heathland immediate threat growth hilversum wetland lies distance compared wetland protection considered highest importance vecht area city hilversum totally grow landscapes case landscapes danger irreversible loss keep wetland safe results presented indicate setting conventional absolute constraint purpose protecting type landscape lead serious problems side effect conclude environmental management project take complex interaction unique spatial arrangement different landscapes region account instead simply putting particular regions protected status according immediate perceived threat day figure third run model equal importance wetland heathland forest details table resulting distribution landscapes shown year intervals table parameters results third run equal importance wetland heathland foresta landscape importance amount remaining lost original amount remaining lost original wetland heathland forest pasture annual growth rate simulation starts year discussion another way protect landscapes threatening urban growth hilversum even effective promoting vertical growth higher buildings instead current trend lateral growth solution efficient protecting valuable landscapes vecht area indication happening near future important insight gained study mode propagation growing city configuration landscapes speed occurs type information useful environmental management projects making easier pick urban areas proximity protected area get included protection plan predicting whether urban area pose significant threat course protection plan another issue importance suggested protection priorities represented model importance values could implemented reality priorities determined remains separate subject issue often exceeds environmental considerations volves factors community demand landscape type recent study kaplan austin community preferences neighboring different landscapes instance shown considerably greater desire forests landscapes regardless benefits might obtained conclusions although urban growth dynamic complex process significant insight still gained experimentation computational models investigation vecht area city hilversum particularly useful demonstrating modeling prediction approach given diversity landscapes present region immediate threat posed urban growth predict pressure protect designated wetland area probably keep city away wetland around expected clash city protection area occur time city would grow valuable landscapes including forests leads suggestion protect wetland also diversity landscapes protection pressure wetland reduced level bringing anticipated clash earlier date still preserving considerable amount wetland way believe even distribution resources achieved still maintaining wetland others important aspect recognize achieved even wetland assigned importance value landscapes certain landscapes change continuously change monitored controlled way beneficial nature human society investigating interplay experiments different protection scenarios study shows even slight adjustment utilization priorities landscapes capable achieving important changes distribution landscapes depending near future references references cordoba generalized gibrat law cities tech rice university houston couclelis cellular automata urban models new principles model development implementation environment planning planning design council europe european landscape convention ets florence geudeke zandvliet eds grote historische atlas van nederland west nederland gilbert van herwijnen lorenz spatial models spatial evaluation analysis wetland restoration vecht river basin regional environmental change glaeser marshall cities information economic growth cityscape journal policy development research henderson handbook economic growth volume north holland amsterdam netherlands urbanization growth idg randstad green heart idg newsletter information documentation centre geography netherlands idg itami simulating spatial dynamics cellular automata theory landscape urban planning jantz goetz shelley using sleuth urban growth model simulate impacts future policy scenarios urban land use metropolitan area environment planning planning design kaplan austin country sprawl quest nature nearby landscape urban planning karen july land use economics study grassland ecological area merced county tech grassland water district los banos kasanko barredo lavalle mccormick demicheli sagris brezger european cities becoming dispersed comparative analysis european urban areas landscape urban planning kusler kentula eds wetland creation restoration status science island press washington lnv natuurbeleidsplan nature policy plan ministerie van landbouw natuur visserij ministry agriculture nature fisheries lnv den haag netherlands lnv ecological networks experiences netherlands ministerie van landbouw natuur voedselkwaliteit ministry agriculture nature food quality lnv den haag netherlands rdg geology netherlands rijks geologische dienst geological survey netherlands rdg haarlem netherlands schiff cellular automata discrete view world hoboken torrens sullivan cellular automata urban simulation environment planning planning design usgs project gigalopolis urban land cover modeling geological survey usgs http accessed may van den bergh barendregt gilbert van herwijnen van horssen kandelaars lorenz spatial economichydroecological modelling evaluation land use impacts vecht wetlands area environmental modeling assessment van eck daalhuizen van den broek van oort raspe randstad network city congress european regional science association ersa land use water management sustainable network society august vrom voortgang verstedelijking vinex vierde nota ruimtelijke ordening extra ministerie volksgezondheid ruimtelijk ordening milieu ministry public housing spatial planning environment vrom den haag netherlands vvv official city guide hilversum central holland vereniging voor vreemdelingen verkeer tourism association netherlands vvv whigham dykyjova hejny eds wetlands world inventory ecology management volume africa australia canada greenland mediterranean mexico papua new guinea south asia tropical south america united states vol handbook vegetation sciences kluwer academic publishers dodrecht netherlands yeh errors uncertainties urban cellular automata computers environment urban systems appendix code model following core code urban growth model used study brevity portion central working model presented leaving auxiliary code interface code compiled executed microsoft framework version namespace hilversumvecht camodel enum celltype heathland urban wetland water canal void rgb reading map presented bitmap bitmap producing bitmap model running modelmapforestcolor fromargb modelmapheathlandcolor fromargb modelmapurbanareacolor fromargb modelmapwetlandcolor fromargb modelmapwatercolor fromargb modelmapcanalcolor fromargb modelmapvoidcolor fromargb celltype modelmap width heig wetlandvalue heathlandvalue abort random rnd camodel width bitmap wetlandvalue heathlandvalue width width igh wetlandvalue wetlandvalue heathlandvalue pasturevalue rnd new random modelmap readmodelmap reads bitmap celltype readmodelmap bitmap map celltype new celltype width hei ght width ight colortocelltype map return bitmap celltype colortocelltype modelmapforestcolor return celltype modelmapheathlandcolor return celltype heathland modelmapwetlandcolor return celltype wetland modelmapurbanareacolor return celltype urban else return celltype void bitmap celltypetocolor celltype celltype return modelmapforestcolor celltype heathland return modelmapheathlandcolor celltype wetland return modelmapwetlandcolor celltype urban return modelmapurbanareacolor celltype void return modelmapvoidcolor else return yellow main method model void run growthrate abort countarea celltype forest countarea celltype wetland countarea celltype heathland countarea celltype urban width ight initialurbanarea initialwetlandarea initialforestarea percentwetlandarea percenturbanarea percentheatharea percentotherarea bitmap map new bitmap width hei ght abort updatemodelmap growthrate countarea celltype countarea celltype wetland countarea celltype heathland countarea celltype urban percentforestarea percentwetlandarea initialwetlandarea percenturbanarea percentheatharea percentotherarea width ight initialotherarea map writemodelmap annual void updatemodelmap growthrate growthrate countarea celltype urban timeout performance timeout rnd next width rnd next ight modelmap celltype urban urbanneighbors modelmap celltype void rnd nextdouble modelmap celltype urban break modelmap celltype rnd nextdouble modelmap celltype urban break modelmap celltype wetland wetlandvalue rnd nextdouble modelmap celltype urban break else heathlandvalue rnd nextdouble modelmap celltype urban break timeout produces bitmap modelmap bitmap writemodelmap bitmap new bitmap width eigh width ight modelmap return counts number urban cell urbanneighbors int modelmap celltype urban modelmap celltype urban modelmap celltype urban modelmap celltype urban modelmap celltype urban modelmap celltype urban modelmap celltype urban modelmap celltype urban return counts number countarea celltype int width ight modelmap return
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types sara capecchi angelo troina dipartimento informatica torino capecchi troina bioambients calculus process algebra suitable representing compartmentalization molecular localization movements compartments paper enrich calculus static type system classifying ambient group types specifying kind compartments ambient stay type system ensures process ambients nested way violates type hierarchy exploiting information given group types also extend operational semantics bioambients rules signalling errors may derive undesired ambients moves merging incompatible tissues thus signal errors help modeller detect locate unwanted situations may arise biological system give practical hints avoid undesired behaviour introduction bioambients variant ambient calculus compartments described hierarchy boundary ambients hierarchy modified suitable operations immediate biological interpretation example interactions compounds reside cytosol nucleus cell could modelled via communications thus bioambients quite suitable representation various aspects molecular localization compartmentalization movement molecules compartments dynamic rearrangement occurs cellular compartments interaction molecules compartmentalized context stochastic semantics bioambients given abstract machine semantics developed bioambients extended operator modelling biomolecular structures applied within dna transcription example technique pathway analysis defined terms static control flow analysis authors apply technique model investigate endocytic pathway facilitates process receptor mediated endocytosis paper extend bioambients calculus static type system classifies ambient group type specifying kind compartments ambient stay words group type describes properties ambients processes group group types defined pairs sets group types intuitively given denotes set ambient groups ambients type stay set ambient groups crossed ambients type one hand set used list elements allowed within compartment complementary elements allowed repelled hand set lists elements cross ambient thus modelling permeability properties compartment starting group types bases define type system ensuring process ambients nested way violates group hierarchy extend operational research funded biobits project converging technologies area regione piemonte merelli quaglia eds biology concurrency back fbtc eptcs capecchi troina work licensed creative commons attribution license types bioambients semantics bioambients exploiting information given group types rules rising warnings signalling errors may derive undesired compartment interactions example correctness capabilities used move compartment inside another compartment checked statically merge capability merges two compartments one capabilities used move compartment inside outside another compartment could cause movement ambient type within ambient type accept cases example incompatible tissues come contact error signal raised dynamically execution system blocked modeller exploit signals helpful debugging information order detect locate unwanted situations may arise biological system intuitively give practical hints avoid undesired behaviour last years growing interest use type disciplines enforce biological properties type system defined ensure wellformedness links protein sites within linked calculus looping sequences see three type systems defined biochemical abstract machine biocham see first one used infer functions proteins reaction model second one infer activation inhibition effects proteins last one infer topology compartments defined type system calculus looping sequences see guarantee soundness reduction rules respect requirement certain elements repellency others finally proposed type system stochastic calculus looping sequences see allows quantitative analysis models presence catalysers inibitors modify speed reactions summary remainder paper organised follows section recall original bioambients syntax section define type system section give typed operational semantics section formulate two motivating examples namely use type system analyse blood transfusions rising errors case incompatible blood types get mixed spore protection bacteriophage viruses finally section draw conclusions bioambients syntax section recall bioambients calculus ambients represent bounded mobile entities nested forming hierarchies provide intuitive mean model compartments components compartment isolated external environment molecular compartments multi molecular complexes molecules partially isolated environment capabilities used model movements changing ambients hierarchies employed model membranes fusion molecules movement complexes formation finally communications model interactions components within across ambients boundaries syntax defined figure difference syntax ambients names optional give name ambient order associate type ambient names ranged channel names ranged capability names ranged use range unspecified names range processes capecchi troina capabilities syncronise using names allow ambient enter sibling ambient accepting enter leave parent ambient exit merge sibling forming unique ambient communications channels prefixed directions denoting different kinds communications local communications loc within ambient sibling communications sts sibling ambients ptc ctp nested ambients concerning processes syntax inaction special case summation denotes process nothing restriction restricts scope name denotes parallel composition stands process replication describes process confined ambient named communication capability choices generalise communication capability prefixes respectively represent standard choices type system classify ambients names group types intuitively type ambient denotes set ambients particular ambient stay describes terms group types possibly including properties ambients processes group group types consist two components form sets group types intuitive meanings types sets following set ambient groups ambients group stay set ambient groups cross may driven respectively enter exit capabilities clearly group type write respectively denote components call guniv type universal environments ambient stay types syntax given figure besides group types capability types type associated name ambients types perform movements described channel types types channels arguments groups capabilities channels notation let capability prefix capability type say either define capability types definition capability type well formed iff none following holds intuitively capability type describing entrance exit ambient type ambient type correct cross stay define environment mapping names types types bioambients loc sts ptc ctp enter accept exit expel actions output input directions parent child child parent capabilities prefixes entry accept exit expel merge merge capabilities processes empty process restriction composition replication ambient communication prefix capability prefix communication choice capability choice figure bioambients syntax capecchi troina group types channels arguments capability types labels channels figure type syntax assume write occur dom dom denotes domain set names occurring environment well formed name associated type well formed following define compatibility group type argument type definition compatibility given capability type group type define compatibility follows iff well formed least one holds check safety bioambients processes using rules figure let type environment derive type names rule typing rules processes shape process set group types representing types ambients set capability types collecting capabilities derives group type empty process indeed empty process stay every type ambient gives parallel composition processes union sets groups obtained typing rule checks whether process safely nested ambient type typed set types ensure every type stay ambient type moreover capability types collected typing must compatible since scope capabilities enclosing ambient capabilities checked admissible emptied rule verifies correspondence type name used capability synchronization capability prefix used adds type rule gives choice union sets groups derived typing show example motivating presence set ambient groups crossed hydrophilic molecules typically capable hydrogen bonding thus enabling dissolve quickly water oil hydrophobic molecules instead tend thus prefer neutral molecules solvents consequence hydrophobic molecules water tend cluster together forming micelles hydrophobic molecules cross cell membranes natural slow way even particular transporter membrane contrary hydrophilic molecules cross membranes dedicated transporters conveyers model crossing properties type system namely represent cells without conveyers ambients type gcconv respectively molecules type phi hydrophilic pho hydrophobic finally transporters type gconv molecules types phi pho stay gcconv cells pho molecules cross cells sets associated types given figure types bioambients figure typing rules group types gcconv gconv phi pho guniv guniv gcconv gcconv gcconv guniv guniv guniv gcconv gcconv figure types molecules cells let cellc gcconv cell conv gconv phi pho phi pho phi pho gconv gconv gcconv gconv gcconv phi pho gconv cell conveyors modeled ambient type gcconv nested conveyors molecules cellc conv enter exit enter expel accept thus model conveyor first accepting molecules exiting current cell entering new cell finally releasing molecules type pho enter inside conveyor finally expelled transport instead molecules type pho also pass membrane cell without use conveyor typed operational semantics section extend semantics bioambients adding rules rise errors consequence undesired behaviour structural congruence bioambients remains unchanged recall capecchi troina figure structural congruence figure rules ambients movements communications model reactions may happen two complementary prefixes name occur parallel safety communications capabilities statically checked typing rules capabilities ensured well formed move ambient type ambient type see definiton hand static control capabilities would require many constraints definition group types check relation group types involved possible interactions consequence type system would restrictive discarding also safe reductions presence potentially unsafe capability prefixes choice reason check reductions signalling errors arise reduction rules figure rule reduces synchronization thorough name enter capability entrance ambient ambient explained rule applied well typed process reduction nesting ambients safe rule reduces synchronization exit prefixes exit ambient ambient put warning parallel new sibling ambients since know ambient arrive exited could nested ambient stay two rules model reduction warned ambients inside another ambient amb reduces warning parallel safe one exit produced unsafe nesting contrary case unsafe nesting amb generates error finally rule models reduction inside ambient warnings rule reduces synchronization prefixes fusion two sibling ambients single one prefix brings processes parallel prefixed one sibling ambient check statically processes parallel prefixed one reduction rule applied perform check runtime raising error case unsafe nesting due merging two sibling ambients rule merge concerning communications rules unchanged model names substitutions due communications processes located ambient ambients parent parent sibling ambients note rules operational semantics checks sets since capability types well formed thus satisfying conditions sets let error range expelerror mergeerror well typed process either reduces another well typed process generates error theorem either error types bioambients enter accept exit expel amb amb expelerror amb mergeerror loc loc merge ptc ctp parent ctp ptc parent sts sts figure operational semantics proof induction definition note semantics reduce warnings top level affect system evolution could useful compositional setting particular entire process nested point another ambient warnings keep conditions admissible ambients without need recompute whole type system motivating examples section provide couple simple motivating examples following use instead assume whenever explicitly represented capecchi troina group types basic elements figure types blood groups blood transfusion example inspired blood type classification blood based presence absence inherited antigenic substances surface red blood cells antigens antigen antigen blood type contains antigens blood type contains antigens blood type contains blood type contains none immune system produce antibodies specifically bind blood group antigen recognized self individuals blood type antibodies individuals blood type antibodies individuals blood type antibodies individuals blood type none antibodies bind antigens surface transfused red blood cells often leading destruction cell reason vital compatible blood selected transfusions another antigen refines classification blood types rhd antigen antigen present blood type called positive else called negative unlike abo blood classification rhd antigen immunogenic meaning person rhd negative likely produce antibodies exposed rhd antigen also common individuals antibodies model blood transfusion system consisting set closed tissues tissues contain blood cells antibodies according classification described join performing transfusion different blood types model red blood cell ambient whose type represents blood type thus groups representing blood types represent rhd antigen antibodies ambients type respectively sets associated different blood types given figure finally model tissue contains red cells ambient type model blood transfusion reduction two tissues complementary merge capabilities donor receiver instance let consider tissue represented ambient type two potential donors types respectively well typed types bioambients figure bacterium could represented membrane containing bacterium dna resistent coated spore represented bacterium surrounded coat bacteriophage depicted outer capsid containing genetic material hypodermic syringe used inject genetic material bacteria cells inject coated cells thus tissue potentially receive blood many donors example human error may also receive blood compatible let consider two possible reductions first one mergeerror results error wrong transfusion causing presence antigen type tissue type second one models transfusion compatible blood types namely bacteriophage viruses section use system model interaction bacteria bacteriophage viruses see figure assume bacterium consists cellular membrane containing dna sporulation mechanism allows producing inactive resistant bacteria forms called spores surrounded membrane coat protecting virus attacks spore germinate produce new bacterium bacterium safely stay ambients containing viruses protected coat types involved model envok envvirus types environments respectively bact gcoat types bacteria protecting membrane gvir type viruses corresponding relevant groups shown figure let consider bacteria surrounded coat omit description dna inside bacterium exit enter expel enter enter well typed bact gcoat gcoat envok gcoat envirus gcoat bact envvirus envok capecchi troina group types basic elements bact gcoat gvir envok envok envvirus envvirus figure example types represent chance bacterium get rid coat capability name allows bacterium germinate exiting protecting membrane coat containing bacterium move every environment bacterium enter envok environments modeled use suitable capabilities put two environments allowing viruses parallel bacterium three possible behaviour following put labels transitions sake readability bacterium gets rid coat enter exit enter expel enter enter accept accept expel enter enter enter accept accept expel enter enter accept accept coat move ambient expel bacterium hostile environment thus generating error exit enter expel enter enter accept accept exit enter expel enter enter accept accept expel enter enter exit enter accept accept expelerror envvirus bact coat move ambient expel bacterium exit enter expel enter enter accept accept exit enter expel enter enter accept accept expel enter enter exit enter accept accept expel enter enter accept enter accept conclusions common approach biologists describe biological systems based use deterministic mathematical means like ode makes possible abstractly reason behaviour types bioambients biological systems perform quantitative silico investigation kind modelling however becomes difficult specification phase analysis processes complexity biological systems taken consideration increases probably one main motivations application computer science formalisms description biological systems motivations also found fact use formal methods computer science permits application analysis techniques practically unknown biologists example static analysis model checking different formalisms either applied inspired biological systems notable models rewrite systems process calculi models advantage allowing direct use many verification tools example model checkers side models based rewrite systems describe biological systems notation easily understood biologists however models rewrite systems compositional possibility study componentwise way behaviour system general naturally ensured process calculi included commonly used describe biological systems paper laid foundations type system bioambients calculus suitable guarantee compatible compartments nesting due intrinsical biological properties framework correctness capabilities checked statically merge capability capabilities could cause movement ambient type within ambient type dynamically rise error used type discipline model incompatible blood transfusion could cause system rise error represent movement bacteria spore friendly environments germinate restart activity acknowledgments would like warmly thank mariangiola encouraged write paper gave crucial suggestions references biocham available http rajeev alur calin belta vijay kumar max mintz hybrid modeling simulation biomolecular networks hybrid systems computation control volume lncs pages springerverlag bogdan aman mariangiola angelo troina type disciplines analysing biologically relevant properties proc mecbic volume entcs pages elsevier roberto barbuti andrea paolo milazzo extending calculus looping sequences model protein interaction proc isbra volume lnbi pages roberto barbuti andrea paolo milazzo paolo tiberi angelo troina stochastic calculus looping sequences modelling simulation cellular pathways transactions computational systems biology roberto barbuti andrea paolo milazzo aangelo troina calculus looping sequences modelling microbiological systems fundamenta livio bioglio mariangiola paola giannini angelo troina type directed semantics calculus looping sequences submitted ijsi linda brodo pierpaolo degano corrado priami stochastic semantics bioambients proc pact volume lncs pages capecchi troina luca cardelli brane calculi interactions biological membranes computational methods systems biology pages springer luca cardelli giorgio ghelli andrew gordon types ambient calculus information computation luca cardelli andrew gordon mobile ambients proc fossacs volume lncs pages mario coppo mariangiola elio giovannetti ivano salvo mobility types mobile processes mobile ambients electronic notes theoretical computer science cats computing australasian theory symposium vincent danos cosimo laneve core formal molecular biology proc esop volume lncs pages mariangiola paola giannini angelo troina type system stochastic cls proc mecbic volume eptcs pages mariangiola paola giannini angelo troina type system elements cls proc dcm volume eptcs pages fages sylvain soliman abstract interpretation types systems biology theoretical computer science cheng zhengwei qiand jinyuan biomolecules modelling bioambients based framework hiroshi matsuno atsushi doi masao nagasaki satoru miyano hybrid petri net representation gene regulatory networks proc psb pages gheorghe membrane computing introduction andrew phillips abstract machine stochastic bioambient calculus proc mecbic volume entcs pages henrik pilegaard flemming nielson hanne riis nielson pathway analysis bioambients journal logic algebraic programming proc nwpt corrado priami paola quaglia beta binders biological interactions computational methods systems biology volume pages aviv regev ekaterina panina william silverman luca cardelli ehud shapiro bioambients abstraction biological compartments theoretical computer science computational systems biology aviv regev ehud shapiro cells computation nature september aviv regev ehud shapiro abstraction biomolecular systems modelling molecular biology pages
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feb complexity various parameterizations common induced subgraph faisal florian abstract maximum common induced subgraph problem henceforth mcis given two graphs one looks graph maximum number vertices induced subgraph mcis among studied classical problems remains many graph classes including forests paper study parameterized complexity mcis generalization clique parameterized size solution even forests structural parameterizations intractable one far parameterizing size minimum vertex cover get tractability indeed parameterized sum vertex cover number two input graphs problem shown tractable algorithm running time log complement result showing unless eth fails solved time log kind tight lower bound shown problems parameters best knowledge vertex cover number also show mcis polynomial kernel parameterized unless finally study mcis connected variant mccis special graph classes respect structural parameters introduction common induced subgraph two graphs graph isomorphic induced subgraph graphs problem finding common induced subgraph maximum number vertices edges many applications number domains including bioinformatics chemistry decision version problem given integer question decide whether solution least vertices say solution size natural parameter problem concerning classical complexity maximum common induced subgraph remains forests common subgraph required connected problem trees moreover maximum common induced subgraph also two input graphs connected bounded treewidth bounded degree particular case maximum common induced subgraph well known induced subgraph isomorphism isi decision problem question posed whether isomorphic induced subgraph words equivalent maximum common induced subgraph case called pattern graph called host graph isi known even interval graph proper interval graph becomes solvable addition extended abstract work appears american university beirut lebanon middlesex university department computer science hendon psl research university cnrs lamsade lebanese london paris france connected unlike subgraph isomorphism induced subgraph isomorphism remains host graph pattern graph consist disjoint union paths parameterized complexity viewpoint problem general natural parameter straightforward reduction therefore mcis also moreover isi therefore mcis remains even graphs interval graphs hand isi fpt nowhere dense graphs expressible formula length function natural parameter generalizes previously known isi free graphs graphs bounded degree observe whenever isi fpt certain graph class mcis see note arbitrary instance mcis reduced instances isi enumerating graph vertices checking whether induced subgraph implies isi mcis parameterized complexity respect natural parameter another way dealing hardness problem study complexity respect auxiliary structural parameters better understand algorithmic behavior see example disjoint union chordless paths mcis hard graphs bounded treewidth well graphs size minimum feedback vertex set bounded thus problem parameterized treewidth input graphs bound sizes minimum feedback vertex sets therefore need look bigger parameters indeed parameter bound sizes minimum vertex covers input graphs problem fpt running time log paper show algorithm significantly improved unless exponential time hypothesis eth fails algorithm solving mcis time log notation suppresses polynomial factors also prove mcis kernel case unless two latter results answer open problems raised finally show maximum common connected induced subgraph mccis solution connected graph also tractable parameterized preliminaries two finite graphs isomorphic bijection given graph graph induced subgraph edge set extremities also say subgraph induced girth graph length shortest cycle contained contracting edge consists deleting replacing vertices single vertex incidence relation edges incident become incident graph minor graph obtained subgraph applying zero edge contractions given fixed graph family graphs said free minor element maximum common induced subgraph problem defined formally follows maximum common induced subgraph mcis input two graphs output induced subgraph isomorphic induced subgraph maximum number vertices maximum common connected induced subgraph mccis defined mcis additional restriction solution must connected completeness also give definition induced subgraph isomorphism induced subgraph isomorphism isi input two graphs output induced subgraph isomorphic exists induced connected subgraph isomorphism icsi defined isi must connected parameterized complexity parameterized problem tractable class fpt respect parameter solved time fpttime computable function constant see details tractability parameterized complexity hierarchy composed classes fpt class contains problems solvable time unrestricted functions problem said even constant value parameter hence problem tractable unless fpt one prove means parameterized reduction problem mapping instance problem time computable function instance function powerful technique design parameterized algorithms kernelization short kernelization algorithm takes instance parameterized problem input computes equivalent instance computable function instance called kernel case function polynomial say polynomial kernel well known problem fpt iff kernel equivalence yields kernels general design efficient parameterized algorithms kernel polynomial even linear size important however lower bounds size kernel shown unless polynomial hierarchy collapses show result use cross composition technique developed bodlaender definition polynomial equivalence relation equivalence relation said polynomial following two conditions hold algorithm given two strings decides whether belong equivalence class time finite set equivalence relation partitions elements classes definition let set let parameterized problem say polynomial equivalence relation algorithm given strings belonging equivalence class computes instance time polynomial bounded polynomial proposition let set karp reductions parameterized problem polynomial kernel unless exponential time hypothesis eth conjecture impagliazzo asserting algorithm instances variables eth together sparsification lemma even implies algorithm solving many algorithmic lower bounds proved eth see example say parameterized problem enumerable feasible solutions enumerated time computable function parameter constant parameterized complexity respect natural parameter study parameterized complexity induced subgraph isomorphism maximum common induced subgraph induced connected subgraph isomorphism maximum common connected induced subgraph respect natural parameter particular study problems graphs bounded degeneracy chordal graphs graphs large girth theorem mcis mccis isi icsi proof since problems straightforward reduction suffices show membership shown problem reduced fpt time simulating turing machine halting steps function turing machine alphabet set states size depending size input initial problem case design turing machine guesses steps corresponding right vertices part necessary right vertices alphabet isomorphic indexing check time whether two induced subgraphs isomorphic connected icsi mccis shown maximum induced bipartite graphs implies mcis bipartite graphs fact show mcis remains restricted graph classes namely bipartite graphs degeneracy particular graphs girth least result tells two things first algorithm cai theorem extended bounded degree bounded degeneracy note problems general graphs become fpt graphs bounded degeneracy dominating set problem second short cycles making hard even without authors present algorithms graphs girth problems general graphs mcis mccis also resistant approach theorem induced subgraph isomorphism induced connected subgraph isomorphism even graphs bipartite graphs degeneracy proof incidence graph graph obtained subdividing edge degeneracy indeed graph bipartite graph edges endpoint vertices degree therefore removed first left independent set transform input instance graphs degeneracy problem consists finding incidence graph within incidence graph show equivalent finding obviously graph isomorphic let assume isomorphic induced subgraph denote vertices degree vertices degree denote isomorphism graph induced subgraph let set every since degree hence degree also every since one looks largest subset vertices induce disjoint union edges two neighbors recall bipartite therefore vertices inducing precisely edges hence membership comes theorem corollary maximum common induced subgraph maximum common connected induced subgraph remain bipartite graphs girth degeneracy absence triangles cycles length four input graphs make problems tractable show absence long induced cycle help either authors show problem dominator coloring fpt input graph chordal specifically four problems chordal graphs fact even show problems remain proper subclass chordal graphs called split graphs split graph graph whose vertex set partitioned set inducing clique independent set theorem isi hence mcis icsi hence mccis remain split graphs proof similarly previous construction define graph edges edges endpoint plus edges graph split induces clique induces independent set instance build equivalent instance soundness obtained way previous proof let say words complexity connected version first note mcis forests mccis solvable case given two forests run mcis algorithm akutsu every pair trees parameterized complexity standpoint maximum common connected induced subgraph fpt whenever induced subgraph isomorphism fpt since enumeration possible induced connected subgraphs used described introduction converse true classes graphs closed adding universal vertex vertex linked vertices instance isi reduced equivalent instance mccis letting graph obtained adding vertex made adjacent vertices structural parameterization let first recall fvs resp fvs represents treewidth resp feedback vertex set number vertex cover number noted parameter combination mcis known even parameter combination fvs fvs bigger combination treewidth problem even since maximum common induced subgraph induced subgraph isomorphism disjoint union chordless paths case parameter equal theorem maximum common induced subgraph parameterized fvs fvs hence one extend result make valid connected version theorem induced connected subgraph isomorphism corollary maximum common connected induced subgraph parameterized fvs fvs proof given instance induced subgraph isomorphism forests least trees build instance induced connected subgraph isomorphism adding universal vertex connected every node graph thus feedback vertex set value one one see two universal vertices must matched together since ones sufficiently high degree solution induced subgraph isomorphism iff solution induced connected subgraph isomorphism result course holds mccis shown mcis fpt parameter combination accordingly problem kernel polynomial bound known size show case kernel polynomial unless theorem unless maximum common induced subgraph polynomial kernel parameterized sum sizes vertex covers two input graphs proof define clique problem given instance tuple question whether graph contains clique vertices given instances gct clique gci graph define equivalence relation strings encoding valid instances equivalent gci gcj equivalent iff gci gcj hereafter assume gci build instance maximum common induced subgraph parameterized vertex cover two graphs vertex cover computed solution size maximum common induced subgraph iff solution size gci describe build build see also figure euv rai euv gci euv euv euv gci euv euv figure illustration construction build see also figure aei figure illustration construction set euv easy see indeed polynomial hence vertex cover size equal size largest instance note size graph depend polynomial size vertex cover also polynomial independent let show induced subgraph iff least one gci clique size suppose gci clique size denote gci clique size exactly gci show induced subgraph size isomorphic set euv one easily check subgraph isomorphic assume induced subgraph denote subgraph isomorphic note triangle pqr indeed bipartite triangle pqr therefore match pqr moreover match since edges besides clique pqr vertex match vertex match vertices euv correspond actual edges gci finally match vertices among note number edges exactly precisely touches edges touches edges order get match one find set edges inducing exactly vertices set vertices clique gci note parameter mcis previous reduction exactly size graphs used proof connected therefore following corollary induced subgraph isomorphism maximum common connected induced subgraph parameterized bound minimum vertex covers input graphs kernel unless algorithm parameter sum vertex cover numbers fact show algorithm unlikely best knowledge first result type parameter vertex cover theorem eth solved time log parameter sum vertex cover number graphs proof give reduction permutation clique linearly preserves parameter known problem admit algorithm running time log unless eth fails permutation clique problem one given graph set vertices goal find clique size row column exactly one vertex part clique row set vertices column set vertices first describe graph built instance permutation clique row resp column index add two vertices resp link edge set rkj resp cjk resp distinguish row indices column indices add clique size link one designated vertex vertices also add clique size special vertex linked vertices finally edge add vertex link four vertices vertex link four vertices ends construction see figure pattern depends defined graph one obtains following construction edges form edges words diagonal nothing else vertex cover size vertices vertices avoid confusion vertices denote vertices sets vertices tilde show reduction valid suppose solution instance permutation clique induced subgraph following mapping map map arbitrary way map raj rbji observe mapping since permutation clique finally map note vertex always exists precisely clique conversely solution instance show permutation one clique size clique size mapped unique vertex degree larger mapped reasons mapped vertices unmatched vertices neighbor vertices matched unmatched vertices neighbor namely similar reasons mapped mapped unmatched vertices exactly one neighbor thus vertices mapped mapped mapped edges resp constrains mapping mapped mapped resp mapped mapped hence see mapping two permutations elements mapped mapped current partial mapping extended solution clique indeed mapped potential vertex exist meaning edge algorithm solving time poly log would therefore translate algorithm running time log permutation clique ignore edges since relevant finding permutation clique figure overall construction represented two edges sake readability edges encoding enhanced distinguish easily edges encoding contradict eth despite fact isi mcis parameterized complexity respect natural parameter exhibit different behaviors considering structural parameters fact latter parameterized vertex cover one two graphs whereas isi fpt parameterized vertex cover second host graph see note host graph vertex cover size minimum size vertex cover pattern graph must bounded parameter otherwise claim follows tractability mcis case given negative result theorem next question pose whether mccis fpt respect size minimum vertex cover parameterized algorithm presented mcis parameter bound minimum vertex cover number input graphs however algorithm help much solving mccis since relies existence feasible solution size least consists mapping two big independent sets two graphs onto course feasible solution mccis following prove mccis tractable theorem maximum common connected induced subgraph parameterized tractable proof time even find minimum vertex covers respectively let independent set assumption parameter max enumerate tripartitions time denote respectively three sets tripartition respectively corresponds vertices matched may deleted comprises vertices matched vertex cover graph vertices matched independent set graph see figure observe partitioned classes twins class twins context set vertices identical neighborhood vertex cover subsets potentially classes empty correspond subset vertex cover exact neighborhood vertex point enumerate mappings time mappings time indeed match vertex vertex twin equivalent thus time enumerate solutions mcis vertices could still matched vertices optimal map independent sets done polynomial time matching greatest number vertices equivalent twin class size smaller two equivalent twin classes twin class equivalent twin class vertices correspondence find solution mccis algorithm described proof enumerates possible maximal common induced subgraphs time current bottleneck improve try match vertices vertex cover vertices independent set connected version problem trivial argument bound size independent set one big trivial solution used connected version used enumeration algorithm mcis corollary maximum common induced subgraph parameterized enumerable let finish section general considerations note isi parameter fvs fvs latter parameter bound vertex cover figure illustration proof theorem dashed boxes represent classes inside independent set arrows represent matching sets vertices sets resp represents vertex cover resp well feedback vertex set makes isi fpt remains open also note isi simple reduction independent set let edgeless graph vertices vertex cover number turn attention case mcis parameterized combination natural parameter structural parameter note general parameterization reduces problem complexity often due tractability mcis free graphs since isi fpt case example consider case parameter sum bound feedback vertex set input graphs natural parameter problem fpt case since graphs vertex set free fixed graph consisting disjoint union triangles applies parameterization treewidth natural parameter considering complete graph vertices example conclusion studied maximum common induced subgraph maximum common connected induced subgraph problems respect solution size special graph classes forests bipartite graphs bounded degree graphs bounded degeneracy graphs graphs without small length cycles two problems tractable free graphs include forests bounded degree graphs bipartite graphs girth degeneracy ruled time two approaches get fixedparameter algorithms subclasses graphs problems considered use structural parameters bound minimum vertex covers input graphs although mcis mccis fpt case proved kernel polynomial bound obtained unless solved time log eth noted mcis even respect smaller auxiliary parameters treewidth feedback vertex set open problems remain addressed example fpt parameterized combination vertex cover number feedback vertex set number vertex cover number treewidth moreover would interesting know whether algorithm mccis theorem improved match lower bound references maximum common induced subgraph parameterized vertex cover inf process bonnet sikora complexity various parameterizations common induced subgraph isomorphism miller froncek editors combinatorial algorithms international workshop iwoca volume lncs pages springer akutsu rnc algorithm finding largest common subtree two trees ieice transactions information systems akutsu polynomial time algorithm finding largest common subgraph almost trees bounded degree ieice transactions fundamentals electronics communications computer sciences alon gutner linear time algorithms finding dominating set fixed size degenerated graphs algorithmica arumugam chandrasekar misra philip saurabh algorithmic aspects dominator colorings graphs iliopoulos smyth editors combinatorial algorithms international workshop iwoca volume lncs pages springer bodlaender jansen kratsch kernelization lower bounds crosscomposition siam discrete cai chan chan random separation new method solving fixedcardinality optimization problems bodlaender langston editors international workshop parameterized exact computation iwpec volume lncs pages springer cesati turing way parameterized complexity comput syst chen kanj xia improved upper bounds vertex cover theor comput damaschke induced subgraph isomorphism cographs proc workshop graph theoretic concepts computer science volume lncs pages springer downey fellows fundamentals parameterized complexity texts computer science springer fellows jansen rosamond towards fully multivariate algorithmics parameter ecology deconstruction computational complexity eur fellows jansen rosamond towards fully multivariate algorithmics parameter ecology deconstruction computational complexity eur flum grohe tractability definability siam journal computing garey johnson computers intractability guide theory freeman san francisco grindley artymiuk rice willett identification tertiary structure resemblance proteins using maximal common subgraph isomorphism algorithm journal molecular biology grohe kreutzer siebertz deciding properties nowhere dense graphs shmoys editor symposium theory computing stoc pages acm heggernes van hof meister villanger induced subgraph isomorphism proper interval bipartite permutation graphs theoretical computer science impagliazzo paturi zane problems strongly exponential complexity journal computer system sciences koch lengauer wanke algorithm finding maximal common subtopologies set protein structures comp biology lokshtanov marx saurabh known algorithms graphs bounded treewidth probably optimal proc annual symposium discrete algorithms soda pages siam lokshtanov marx saurabh slightly superexponential parameterized problems proc annual symposium discrete algorithms soda pages siam marx schlotter cleaning interval graphs algorithmica mcgregor willett use maximal common subgraph algorithm automatic identification ostensible bond changes occurring chemical reactions chemical information computer science moser sikdar parameterized complexity induced matching problem discrete applied mathematics niedermeier invitation fixed parameter algorithms lecture series mathematics applications oxford university press raman saurabh short cycles make problems hard fpt algorithms problems graphs short cycles algorithmica raymond willett maximum common subgraph isomorphism algorithms matching chemical structures journal molecular design yamaguchi aoki mamitsuka finding maximum common subgraph partial graph polynomially bounded number spanning trees inf process
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proc eur conf python science euroscipy bloscpack compressed lightweight serialization format numerical data valentin apr paper introduces bloscpack file format accompanying python reference implementation bloscpack lightweight compressed binary based blosc codec designed lightweight fast serialization numerical data article presents features api aspects reference implementation particular ability handle numpy ndarrays furthermore order demonstrate utility format compared alternative lightweight serialization solutions numpy ndarrays performance comparisons take form comprehensive benchmarks range different artificial datasets varying size complexity results presented last section article index information theory python numpy file format serialization blosc ntroduction using compression storage numerical data two potential improvements one make first using compression naturally one save storage space often save time using compression serialization total compression time sum time taken perform compression time taken write compressed data storage medium depending compression speed compression ratio sum maybe less time taken serialize data uncompressed format writeuncompressed writecompressed timecompress bloscpack file format python reference implementation aims achieve exactly leveraging fast multithreaded blocking shuffling blosc codec losc blosc blosc fast multitreaded blocking shuffling compressor designed initially compression contrary many available compressors operate sequentially data buffer blosc uses blocking technique split dataset individual blocks operate block using different thread effectively leads multithreaded corresponding author valentin independent copyright valentin haenel article distributed terms creative commons attribution license permits unrestricted use distribution reproduction medium provided original author source credited http compressor block size chosen either fits typical cache compression levels cache compression levels larger modern cpus typically cores choice block size leads optimal performance operation also blosc features shuffle filter may reshuffle elements byte doubles significance net result series numerical elements little difference elements close similar bytes placed closer together thus better compressed specially true time series datasets internally blosc uses codec blosclz derivative fastlz fastlz implements scheme reason blosc introduce codec mainly desire simplicity blosclz highly streamlined version fastlz well providing better interaction blosc infrastructure moreover blosc designed extensible allows codecs blosclz used words one consider blosc handles splitting data blocks optionally applying shuffle filter future filters responsible coordinating individual threads operation blosc relies real codec perform actual compression data blocks one think blosc way parallelize existing codecs allowing apply filters also called fact time research presented paper conducted summer implementation existed integrate well known snappy codec snappy well blosc framework january proof concept matured version blosc comes equipped support snappy snappy even zlib zlib blosc initially developed support compression order mitigate effects memory hierarchy specifically mitigate effects memory latency ever growing divide cpu speed memory access also known problem starving cpus goal compression techniques numerical container keeps data compressed blocks data needs operated decompressed caches cpu hence data proc eur conf python science euroscipy moved faster memory cpu net result faster computation since less cpu cycles wasted waiting data similar techniques applied successfully settings imagine example one wishes transfer binary files internet case transfer time significantly improved compressing data transferring decompressing received result total compressed transfer time taken sum compression decompression process time taken transfer compressed file less time taken transfer plain file example well known unix tool rsync rsync implements switch performs compression data sending decompression receiving basic principle applies compression except transferring data memory cpu initial implementations based blosc exist blaze blaze carray carray shown yield favourable results personal communication francesc alted bloscpack format reference implementation builds serialization format around blosc codec simple chunked well suited storage numerical data described bloscpack format description follows umpy numpy numpy ndarray multidimensional numerical container scientific python applications probably fundamental package scientific python ecosystem widely used relied upon libraries applications consists array class various different initialization routines many different ways operate data efficiently xisting ightweight olutions number plain uncompressed compressed lightweight serialization formats numpy arrays compare bloscpack specifically ignore heavyweight solutions comparison npy npz zfile npy npy npy simple plain serialization format numpy considered somewhat gold standard serialization one advantages lightweight format specification simple easily digested within hour essence simply contains ndarray metadata serialized data block metadata amounts dtype order shape array main drawback plain serialization format support compression npz npz simply put zip file contains multiple npy files since zip file may optionally compressed however main uses case store multiple ndarrays single file zip implementation deflate deflate algorithm unlike evaluated compressed formats npz support compression level setting zfile zfile native serialization format ships joblib joblib framework joblib equipped caching mechanism supports caching input output arguments functions thus avoid running heavy computations input changed serializing ndarrays joblib special subclass pickler used store metadata whereas datablock serialized zfile zfile uses zlib zlib internally simply runs zlib entire data buffer zlib also implementation deflate algorithm one drawback current zfile implementation chunking scheme employed means memory requirements might twice original input imagine trying compress incompressible buffer case memory requirement would since entire buffer must copied memory part compression process written disk loscpack ormat format contains byte header contains various options settings file example magic string format version number total number chunks meta section variable size contain metadata needs saved alongside data optional offsets section provided allow partial decompression file future followed series chunks blosc compressed buffer chunk optionally followed checksum compressed data help protect silent data corruption chunked format initially chosen circumvent limitation blosc codec fact zfile format suffers exact limitation since least python also limited buffers size limitation stems fact used internally algorithms store size buffer maximum value indeed case using chunked scheme turned useful right using modest current default causes less stress memory subsystem also means contrast zfile small fixed overhead equal required compression decompression process example compressing decompression external storage medium version format enhanced allow appending data existing bloscpack compressed file achieved offsets metadata section dummy values allow chunks appended later metadata enlarged one caveat bloscpack compressed lightweight serialization format numerical data infinite amount space limited amount data potentially appended however provide potential consumers format much flexibility possible amount space configurable discussion technical details bloscpack format interested reader advised consult official documentation bloscpack contains full description header layout sizes entries permissible values ommand ine nterface initially bloscpack conceived compression tool time writing python api development fact interface used drive python api contrary existing tools gzip gzip bloscpack use options control mode operation instead uses subcommand style simple example compress decompress another interesting subcommand info used inspect header metadata existing file info bloscpack documentation contains extensive descriptions various options many examples use command line api packing umpy rrays version bloscpack comes support serializing numpy ndarrays approach simple lightweight data buffer saved blosc compressed chunks defined bloscpack format shape dtype order ones saved npy saved metadata section upon first empty ndarray allocated information three metadata attributes bloscpack chunks decompressed directly array bloscpack python api numpy ndarray similar simple npy interface arrays using single function invocations example serializing numpy array file import numpy import bloscpack assert example serializing string import numpy import bloscpack assert compression parameters configured keyword arguments pack functions see documentation detail omparison npy npy specification lists number requirements npy format compare npy bloscpack featurewise let look extent bloscpack satisfies requirements dealing numpy ndarrays represent numpy arrays including nested record arrays object arrays since support numpy ndarrays fresh empirical results using toy arrays tested simple integer floating point types string arrays seem work fine structured arrays also supported even nested data types finally object arrays also seem survive tests represent data native binary form since bloscpack compress data impossible represent data native binary form contained single file using metadata section bloscpack format required metadata decompressing numpy ndarray included alongside compressed data support arrays directly array fortran ordering save fortran ordering bloscpack order saved part metadata contiguous memory block saved order set decompression hence array deserialized correctly store necessary information reconstruct array including shape dtype machine different architecture endianness type mentioned integer types well string object arrays handled correctly shape preserved endianness initial toy examples dtypes pass roundtrip test reverse engineered case reverse engineering refers ability decode bloscpack compressed file bloscpack code specification lost npy achieved one roughly knows look namely three metadata attributes one plain data block bloscpack case things difficult first header larger number entries must first deciphered secondly data compressed without knowledge compression scheme implementation difficult reverse engineer allow data proc eur conf python science euroscipy since data compressed possible use mmap primitive map file memory meaningful way however due chunkwise nature storage theoretically possible implement scheme using chunk offsets typesize shape numpy ndarray metadata possible determine chunk chunks contain single element range thus load decompress chunks disk read stream object instead actual file part bloscpack code base since early versions since essential unit testing touching file system using stringio object fact numpy ndarray strings implemented read written using apis provided numpy package bloscpack explicitly aspire part numpy timing timing algorithm used modified version timeit tool included python standard library supports deactivation python interpreters garbage collector run executing code run example measuring decompression speed without linux file system cache one needs clear cache run imperative operation enter timing also measuring compression speed one needs make sure sync executed run ensure data actually written storage medium contrary clearing file system cache time required sync operation must enter timing contaminate results enchmarks benchmarks designed compare following three alternative serialization formats numpy ndarrays npy npz zfile bloscpack end measured compression speed decompression speed without linux file system cache compression ratio number different experimental parameters minimum across sets taken final value datapoint small size sets runs performed mid size sets runs performed finally large size sets runs performed parameters three different array sizes chosen small bytes mid bytes large bytes three different dataset complexities chosen low arange low kolmogorov complexity medium sin noise high random numbers lastly two different storage mediums chosen ssd encrypted luks ssd card card chosen represent class slow storage actually expect serialize anything card practice cut number data points choose evaluate compression levels zfile bloscpack although npz compressed format support modifying compression level results using different codec values configuration leads data points additionally account fluctuations datapoint run multiple times depending size dataset case number sets number runs performed mean across runs set hardware machine used lenovo carbon ultrabook intel core processor cpu processor physical cores active hyperthreading resulting threads cpu scaling governor set performance resulted cpu frequency per core cpu three levels cache reported linux sysfs memory bandwidth reported write read blosc benchmarking tool interestingly stark contrast reported maximum memory bandwidth advertised manufacturers data sheet operating system used debian stable following kernel installed debian backports smp debian bandwidth two storage media benchmarked using ssd write read inquisitive reader note following caveat stage perhaps kolmogorov complexity correct choice complexity measure define low entropy data style dictionary encoder fact sequence consecutive integers definition high lempelziv complexity compressible however shown benchmarks later bloscpack actually good compressing kinds sequences whereas zfile npz result fact arange generated type integer data shuffle filter bloscpack optimize well stage simply state proposed low entropy dataset sufficient benchmarks treatment effects shuffle filter complexity beyond scope paper perhaps subject future publication bloscpack compressed lightweight serialization format numerical data disabled defaults additionally certain features operating system disabled explicitly running benchmarks optimizations chosen based empirical observations running initial benchmarks observing suspicious behaviour investigating possible causes may operating system effects precautions listed next found observably detrimental effects disabling lead increased reliability results first daily cronjobs disabled commenting corresponding line important running benchmarks night certain intensive cronjobs might contaminate benchmarks secondly laptop mode tools disabled via setting tools regulate certain resource settings particular disk latency cpu frequency scaling governor certain system connectivity might contaminate benchmarks ersions sed following versions used acquire data reported article bloscpack joblib numpy conda python python anaconda results files available repository talk bloscpack results file analysed contained csv file bloscpack settings order reduce overhead running bloscpack optional features enabled benchmarks particular checksum used compressed chunks offsets chunks stored esults results benchmark presented figures figures show timing results collection subplots subplot shows timing results given combination dataset size entropy dataset size increases horizontally across subplots whereas dataset entropy increases vertically across subplots figures show results ssd storage type figures four show results storage type figures compare bloscpack npy whereas figures compare bloscpack npz zfile npy shown separately npz zfile since performance characteristics different adequately compared visually plot timing plots black bars indicate compression time white used denote decompression time file system cache gray identifies decompression time hot file system cache timing plots larger values indicate worse performance lastly figure shows compression ratios examined formats fig see bloscpack compares npy ssd storage type first thing note small datasets first column subplots bloscpack lag behind much compared npy compression actually slightly faster decompression however absolute differences millisecond range one might perhaps argue bloscpack npy par small datasets soon move medium size datasets first gains seen especially low entropy case bloscpack beats npy compression decompression file system cache medium entropy case bloscpack slightly faster settings least compression decompression cases surprisingly decompression hot file system cache bloscpack actually times slower compression levels one possibility might even though file contents memory reading file necessitates initial copy data actually decompressed high entropy case bloscpack mostly slightly slower npy large dataset results simply scaled version medium dataset size results yield additional insights fig shows comparison bloscpack npz zfile ssd storage type comparison speed blosc compressor really shines every combination dataset size entropy compression level bloscpack compress faster competitors extreme case large size low entropy bloscpack times faster compression npz seconds npz seconds bloscpack even high entropy case little compression possible due statistics dataset bloscpack significantly faster compression presumably blosc try compress buffer finish quickly work done simply copies input verbatim one surprising result npz zfile level take extraordinary amounts time compress low entropy dataset fact take longest low entropy dataset compared medium high entropies potentially related high complexity dataset mentioned recall npz zfile use deflate algorithm belongs class dictionary encoders may suffer since shuffle filter case blosc employed figures show results figures respectively storage class since card much slower ssd card task strongly bound therefore benefits compression reaped earlier example bloscpack level twice fast npy compression medium size medium entropy dataset low entropy dataset medium large sizes bloscpack proc eur conf python science euroscipy order magnitude faster high entropy dataset bloscpack par npy overhead trying compress succeeding negligible due boundedness resulting speed card comparing bloscpack npz zfile card boundedness means algorithm achieve high compression ratio reasonable amount time perform better example medium size medium entropy npz actually times faster bloscpack compression ssd case observe npz zfile perform slowly low entropy data lastly figure see compression ratios codec size entropy mostly sanity check npy always since plain serialization format bloscpack gives better compression ratios low entropy data npz zfile give better compression ratios medium entropy data serializers give ratio close zero high entropy dataset onclusion article introduced bloscpack python reference implementation features file format presented compared serialization formats context numpy ndarrays benchmarking results presented show bloscpack yield performance improvements serializing numpy arrays compared existing solutions variety different circumstances uture ork results obtained far open questions remain unsolved first clear bloscpack level gives comparatively bad results decompressing hot file system cache also bad performance zfile npy low entropy dataset must investigated perhaps alternative found biased towards bloscpack additionally mathematical insights complexity reduction properties blosc shuffle filter would valuable lastly comprehensive benchmarks need run means first finding benchmark datasets establishing corpus run bloscpack solutions furthermore would nice run benchmarks architectures machines physical cores memory access nfs commonly found compute clusters ratitude author would like thank following people advice helpful comments discussions pauli virtanen varoquaux robert kern philippe gervais also author would like specially thank van der walt francecs alted reviewing drafts paper eferences francesc alted data access problem euroscipy keynote presentation http francesc alted modern cpus starving done computing science engineering vol march http deflate peter deutsch deflate compressed data format specification version http valentin haenel introducing bloscpack euroscipy presentation https bruce jacob memory system avoid ignore fake synthesis lectures computer architecture pages stefan van der walt chris colbert varoquaux numpy array structure efficient numerical computation computing science engineering ziv jacob lempel abraham may universal algorithm sequential data compression ieee transactions information theory npy robert kern npy format https joblib joblib http zlib zlib http gzip gzip http rsync rsync http blaze blaze http carray carray http numpy numpy http fastlz fastlz http snappy snappy http http blosc blosc http bloscpack bloscpack https cpu processor bloscpack compressed lightweight serialization format numerical data small size low entropy medium size low entropy large size low entropy npy bloscpack bloscpack codec level codec level codec level bloscpack compress decompress bloscpack small size medium entropy medium size medium entropy large size medium entropy npy bloscpack bloscpack codec level codec level codec level bloscpack bloscpack small size high entropy medium size high entropy large size high entropy npy bloscpack bloscpack codec level codec level codec level bloscpack bloscpack fig compare bloscpack npy ssd storage type proc eur conf python science euroscipy small size low entropy medium size low entropy large size low entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile compress decompress bloscpack bloscpack small size medium entropy medium size medium entropy large size medium entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile bloscpack bloscpack small size high entropy medium size high entropy large size high entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile bloscpack bloscpack fig compare bloscpack npz zfile ssd storage type bloscpack compressed lightweight serialization format numerical data small size low entropy medium size low entropy large size low entropy npy bloscpack bloscpack codec level codec level codec level bloscpack compress decompress bloscpack small size medium entropy medium size medium entropy large size medium entropy npy bloscpack bloscpack codec level codec level codec level bloscpack bloscpack small size high entropy medium size high entropy large size high entropy npy bloscpack bloscpack codec level codec level codec level bloscpack bloscpack fig compare bloscpack npy storage type proc eur conf python science euroscipy small size low entropy medium size low entropy large size low entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile compress decompress bloscpack bloscpack small size medium entropy medium size medium entropy large size medium entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile bloscpack bloscpack small size high entropy medium size high entropy large size high entropy zfile zfile npz bloscpack bloscpack codec level codec level codec level zfile bloscpack bloscpack fig compare bloscpack npz zfile storage type bloscpack compressed lightweight serialization format numerical data codec level size entropy size entropy codec level size entropy zfile zfile zfile npy npz bloscpack bloscpack bloscpack bloscpack size entropy codec level size entropy codec level size entropy codec level size entropy codec level zfile zfile zfile npy npz bloscpack bloscpack bloscpack bloscpack size entropy codec level codec level zfile zfile zfile npy npz bloscpack bloscpack bloscpack bloscpack codec level size entropy fig compression ratios examined formats proc eur conf python science euroscipy
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bound index quadratic form scalar extension function field quadric jul stephen scully abstract let anisotropic quadratic form defined general field article formulate new upper bound isotropy index scalar extension function field arbitrary quadric one hand bound offers refinement celebrated bound established earlier work totaro direct generalization karpenko theorem possible values first higher isotropy index prove validity two important cases case char case char quasilinear diagonalizable two cases treated separately using completely different approaches first second purely algebraic introduction basic yet fundamental tool study quadratic forms general fields scalar extension among many uses perhaps significant forcing quadratic form interest acquire zero assuming none exist base field forcing anisotropic form become isotropic effect lowering anisotropic dimension form apart providing natural means argue inductively often lead one witness behaviour quadratic forms interest field definition enlarged order draw meaningful conclusions challenge typically one determining restrictions general nature kind behaviour restrictions however depend heavily particular choice extension field order maintain much control possible often desirable choose extension restrictions severe philosophically means one choose extension sense valuation theory generic problem hand purpose forcing anisotropic quadratic form become isotropic canonical choice extension property namely function field projective quadric defined vanishing form reasons studying effect scalar extension function fields quadrics dominant theme much research carried algebraic theory quadratic forms since early one problem central interest area following let anisotropic quadratic forms dimension general field let denote projective defined vanishing respectively circumstances become isotropic function field quadric perspective simply amounts asking necessary sufficient conditions order exist rational map nevertheless fact impose constraints triple endows problem depth complexity belies initial appearance fact mathematics subject classification key words phrases quadratic forms isotropy indices quadrics quadratic grassmannians rational maps quadratic grassmannians stephen scully entirely unreasonable hope complete solution generality vast literature accumulated past three decades amply demonstrates problem already considerably involved situations see present article thus concerned weaker variant problem seeks investigate restrictions particular kind isotropy index scalar extension field recall quadratic form defined vectors space field isotropy index defined maximal dimension subspace uniformly zero totally isotropic subspace integer detect isotropy measure extent persists language algebraic geometry problem concerned identifying necessary conditions order exist rational map variety totally isotropic subspaces prescribed dimension restrictions interested general nature namely imposed basic discrete invariants respect problem long rich history dating back late classic subform theorem cassels pfister many important contributions made years large number authors one notable results established date direction penetrating upper bound integer involving three invariants dimension dimension first higher isotropy index defined integer originally conjectured izhboldin origin seminal work hoffmann bound first established case char karpenko merkurjev cor later full generality totaro see thm theorem totaro let anisotropic quadratic forms dimension field isotropic dim dim unlike much work came theorem established using methods algebraic geometry indeed sets authors exploited aforementioned geometric interpretation integer order bring theory algebraic cycles bear problem built upon foundational ideas vishik earlier obtained results direction theorem significant due fact considerable amount known concerning integer fact precise conjectural description possible values due hoffmann conjecture verified case char karpenko unfortunately karpenko proof makes essential use powerful fact yet available characteristic existence operations chow groups smooth projective varieties coefficients hoffmann conjecture therefore remains open latter setting recent work author however completely different purely algebraic methods used settle conjecture special class forms characteristic specifically shown thm char hoffmann conjecture holds case quasilinear meaning isometric form fermat type weighted sum squares left following general result theorem karpenko scully let quadratic form dimension field let dim assume either characteristic characteristic quasilinear notation stands order integer worth remarking hoffmann conjecture essentially wide open forms characteristic partial results obtained also haution efforts develop geometric machinery currently absent setting see theorems represent important landmarks theory quadratic forms put together lead highly applicable conclusion typically much dim dim geometrically speaking may interpreted concretely fact anisotropic quadric rationally compressed another quadric sufficiently lower dimension striking behaviour already observed aforementioned work hoffmann proved less refined variant theorem integer dim replaced largest power strictly less dim least char see noted observation since gone great influence developments closely related topics within theory algebraic groups see main aim current article present new upper bound integer one hand complementary direct generalization hoffmann conjecture possible values first higher isotropy index strengthens unifies important part existing literature per theorem however must limit considerations quasilinear forms working setting nevertheless results obtain give firm indication bound hold without restriction main theorem thus following reader readily observe theorem nothing else special case theorem let anisotropic quadratic forms dimension field let dim assume either characteristic characteristic quasilinear max dim dim although restrict quasilinear forms characteristic worth stressing proof case theorem author mind striking part paper furthermore contrary traditional order events characteristic case theorem fact preceded quasilinear case indeed stated bound discovered way conceptual trivialization established quasilinear setting effective advertisement study quasilinear quadratic forms intrinsic interest aside quasilinear case sometimes serve guiding light investigations general theory give another illustration unifying power theorem show may used give short proof deep result characteristic originally conjectured knebusch proven fitzgerald contrast result follow directly theorems analogous result holds quasilinear forms characteristic see prop may also deduced theorem similar way stephen scully example fitzgerald see thm let anisotropic quadratic forms dimension characteristic becomes hyperbolic dim dim dim similar pfister form proof fitzgerald original proof involved rather subtle manipulations casselspfister subform theorem thm among standard results classical algebraic theory quadratic forms given theorem however statement follows straightforward manner indeed since dim dim dim hypothesis theorem tells case dim dim becomes hyperbolic subform theorem implies dim dim definition therefore dim dim suppose similar pfister form exists extension isotropic anisotropic kernel similar pfister form example take penultimate entry generic splitting tower see becomes hyperbolic another application subform theorem together shows dim since similar pfister form dimension equal power possible thus conclude similar pfister form desired characteristic case theorem proved fact get slightly refined assertion see theorem argument makes use theory chow correspondences similar way proofs aforementioned results karpenko though present using conceptual language chow motives rely crucially recent advance study motivic decompositions quadrics due vishik since vishik work involves heavy use aforementioned steenrod operations chow groups modulo approach theorem certainly limited characteristic present order treat case char quasilinear therefore forced proceed entirely different manner follow pattern ideas developed primary theorem main result setting concerns certain structural decomposition anisotropic kernel form see theorem result may viewed generalization thm takes place level quadratic forms quasilinear part theorem emerges nothing consequence validity actually get slightly better statement see corollaries interestingly quasilinear cases theorem also immediate consequences theorem additional new results seem known analogues characteristic theory conjectural otherwise see therefore left conceptually satisfying unification several important statements setting old new taking account would interesting know whether algebraic approach isotropy problem theorem particular admits kind generalization beyond quasilinear case convenience reader proofs two cases theorem preceded separate section preliminaries particular relevant situation definitions basic concepts results common given next section finally remark proofs main results quasilinear quadratic forms readily generalize give analogous results quasilinear prime degree forms degree characteristic sake transparency however refrain working generality terminology text word scheme means scheme field word variety means integral scheme diagonalizable quadratic forms let field space quadratic form mean map associated polar form given map dual space given bijective say present article concerned quadratic forms diagonalizable sense arise diagonal part symmetric bilinear form field definition precisely said diagonalizable exists symmetric bilinear form char every quadratic form property indeed must take case contrast char diagonalizable quasilinear discussion concern entire theory quadratic forms fields characteristic quasilinear part theory terminology section quadratic form simply form shall mean diagonalizable quadratic form space recall basic concepts results concerning diagonalizable quadratic forms arbitrary fields reader referred comprehensive discussion basic concepts let quadratic form space defined denoted dimension vector space called dimension denoted dim set consisting elements represented denoted given field extension unique quadratic form space extends whose polar form extends denote form subring containing denote subset given write quadratic form vector space let another quadratic form exists bijective map say isometric write say similar element said similarity factor set similarity factors evidently subgroup write union note assignment defines quadratic form called orthogonal sum given positive integer denote stephen scully orthogonal sum copies note exists quadratic form say subform tensor product unique quadratic form polar form given kronecker product maps element exists quadratic form say divisible write quadratic form pngiven elements space every quadratic form sense considering clearly isometric one type vector said isotropic respect subspace consisting entirely isotropic vectors said totally isotropic say isotropic contains isotropic vector anisotropic otherwise equivalently anisotropic projective quadric points char implies smooth prop contrast char totally singular sense smooth points irrespective whether anisotropic isotropy index denoted defined maximal dimension totally isotropic subspace char commonly known witt index characteristic integer insensitive making purely transcendental extensions base field see lem decomposition isotropic forms quadratic form one may associate anisotropic quadratic form qan known anisotropic part precise nature form depends characteristic char assignment defines quadratic form vector space known hyperbolic plane isometry unique isotropic form dimension case witt decomposition theorem see thm characterizes qan unique anisotropic form isometry qan note particular dim qan dim situation char quasilinear additivity implies set isotropic vectors fact subspace particular isotropy index nothing else dimension case form qan may defined restriction quotient space isometry qan unique anisotropic form qan another point view additivity also implies set consisting elements represented subspace form qan may also characterised isometry unique anisotropic form qan see proposition important note setting dim qan dim opposed aforementioned dim qan dim prevails nondegenerate forms characteristic nevertheless characteristics see dimension qan measures extent isotropic dim qan say split note algebraically closed every quadratic form split moreover char one requires perfect order make conclusion example forms finite field characteristic function fields quadrics let quadratic form split irreducible element symmetric algebra projective quadric integral affine cone see case write function field former latter field course purely transcendental extension field extension simply write instead whenever defined note frac systems algebraically independent variables evidently form isotropic knebusch splitting pattern let quadratic form knebusch introduced following construction least case char set qan recursively define provided split qfr provided defined note defined dim dim since anisotropic quadratic form becomes isotropic function field quadric defines knebusch process finite terminating first integer split integer called height tower fields called knebusch splitting tower set qfr split integer called higher isotropic index denoted case form called higher anisotropic kernel char tower usually referred generic splitting tower generic sense among towers extensions ultimately split particular field extension exists case integer qfr easily deduced following specific statement lemma see let quadratic form field characteristic knebusch splitting tower let integer field extension free compositum purely transcendental extension lemma follows fact smooth projective quadric admits rational point rational variety prop case totally singular quadrics see rem iii whence characteristic restriction pfister forms let positive integer given write quadratic form defined tensor product char forms type known pfister forms char commonly referred nfold forms order distinguish symmetric bilinear forms also bear pfister name either case form round multiplicative meaning char round anisotropic forms prop contrast char stephen scully anisotropic round forms case anisotropic pfister forms precisely anisotropic forms strongly multiplicative sense every field extension see theorem preliminaries characteristic begin investigations case fields characteristic different make use certain interplay exists setting knebusch splitting pattern anisotropic quadratic form discrete motivic invariants associated quadric therefore begin recalling basic concepts results concerning interaction remainder section denote arbitrary field characteristic motivic decomposition type upper motives readable introductions theory chow motives particular emphasis motives quadrics reader referred field write chow additive category motives integral coefficients defined example smooth projective variety write denote motive element chow special case spec simply write instead thus suppressing dependency base field given integer object chow write tate twist particular denote tate motive shift chow field extension write denote image object chow natural scalar extension functor chow chow let smooth projective quadric dimension fixed field characteristic result vishik see direct summand decomposes finite direct sum indecomposable objects chow furthermore decomposition unique reordering summands case split vanishing locus split quadratic form full decomposition obtained rost showed decomposes certain direct sum tate motives since every smooth projective quadric algebraically closed field split follows decomposes direct sum tate motives chow denotes fixed algebraic closure precise statement following odd even see prop indecomposable direct summand follows uniquely decomposes direct sum subset tate motives appearing decomposition moreover subsets arise distinct indecomposable summands necessarily disjoint see complete motivic decomposition therefore determines partition tate motives appear partition important discrete invariant known motivic decomposition type motivic decomposition type interacts nontrivially known discrete invariants particular interacts knebusch splitting pattern underlying form interaction exploited obtain deep results concerning latter invariant recent years order prove characteristic part theorem need recent advance study motivic decomposition type anisotropic quadrics result due vishik first following define upper motive unique indecomposable direct summand trivial tate motive isomorphic direct summand vishik result gives following information regarding anisotropic case recall anisotropic quadric characteristic necessarily smooth see theorem vishik see thm let anisotropic quadratic form dimension associated smooth projective quadric write dim uniquely determined integers set even odd tate motive isomorphic direct summand also make use following elementary observation also due vishik relates two anisotropic quadrics motivic level situation one quadrics stably birational variety totally isotropic subspaces prescribed dimension see also thm strengthening result theorem vishik prop thm let anisotropic quadratic forms dimension associated smooth projective quadrics respectively suppose every field extension isomorphic direct summand criterion stable birational equivalence smooth projective quadrics let anisotropic quadratic forms dimension isotropic see dim dim application theorem order make use theorem situation theorem need following complementary statement also due karpenko merkurjev theorem see thm let anisotropic quadratic forms dimension isotropic dim dim isotropic well remark totaro showed theorem also valid characteristic irrespective smoothness assumptions shall need result main results characteristic case section show theorem valid characteristic following previous section denote arbitrary field characteristic throughout note anisotropic quadratic form dimension theorem asserts integer dim characteristic case theorem therefore follows following refined statement theorem let anisotropic quadratic forms dimension let dim max dim dim stephen scully proof order simplify notation set assertion reads max dim dim idea proof following let necessarily smooth projective defined vanishing respectively assuming fails holds show modulo inductive assumption certain shift upper motive direct summand motive see show possible using theorem analyse motivic decomposition types key observation following lemma suppose situation hold field extension proof let knebusch splitting tower let higher anisotropic kernel forms see recall integer denotes isotropy index qfk since fails hold hypothesis particular isotropic thus genericity knebusch construction exists integer see claim isotropic order prove claim may clearly assume pfr anisotropic fact one show indeed case need know working assumption let set dim dim pfr pfr since becomes isotropic definition karpenko merkurjev theorem shows prove claim suffices check remarks preceding theorem certainly see equality holds let first note purely transcendental extension lemma since isotropy indices insensitive purely transcendental extensions follows pfr particular dim dim since dim dim hypothesis see dim dim dim dim dim dim hand since also assuming karpenko theorem see theorem follows dim divisible integer satisfying dim dim dim definition divisible follows divisible min since min shows thus proves initial claim isotropic finally complete proof let field extension lemma purely transcendental extension conversely lemma shows compositum purely transcendental extension conditions therefore equivalent wanted prove return proof theorem argue induction dim case dim trivial assume dim theorem valid forms dimension dim inductive hypothesis show valid use following trivial observation lemma let quadratic form let field extension integer exists subform dim dim proof enough treat case argue case induction dim case dim trivial assume dim let totally isotropic subspace dimension hyperplane intersection least particular subform induction hypothesis admits subform dimension dim since subform result therefore follows suppose sake contradiction inequality valid lemma exists subform codimension least statement theorem fails hold pair inductive assumption therefore conclude since fails hold lemma shows condition theorem satisfied hence isomorphic direct summand note however isomorphic upper motive indeed since trivial tate motive isomorphic direct summand thus isomorphic distinct indecomposable direct summands complete proof obtain contradiction initial supposition showing common tate motive respective decompositions happen direct summands see use vishik theorem specifically use vishik result show isomorphic direct summand dim former case follows immediately theorem indeed reader quickly check case latter result states precisely isomorphic direct summand see isomorphic direct summand however first need analyse alternating integer dim specifically need following observation lemma suppose situation write dim uniquely determined integers exists even integer stephen scully given assertion obtain isomorphic direct summand applying theorem case proving lemma make intermediate calculation sublemma situation integer dim dim positive proof let knebusch splitting tower let see since assuming hold proof lemma shows dim dim dim dim since claim therefore amounts assertion suppose sake contradiction case hand since becomes dim dim karpenko theorem see theorem definition thus conclude supposition incorrect claim valid proof lemma begin observing write dim dim dim dim sublemma positive hand since fails hold integer dim dim strictly less since conclude definition dim unique integers adding see even dim odd irrespective whether odd even follows dim even integer integers time since write uniquely determined integers dim precisely description dim alternating sum appears statement lemma since dim definition done lemma proved shown indecomposable direct summands admit common tate motive respective decompositions scalar extension thus providing needed contradiction completes induction step theorem proved remark theorem indeed refinement theorem characteristic setting example integer negative theorem get slightly better result case preliminaries characteristic turn case quasilinear quadratic forms characteristic begin collecting preliminary results used later detailed expositions basic theory quasilinear quadratic forms reader referred remainder section denote arbitrary field characteristic representation theorem well known anisotropic quadratic form arbitrary field recovered isometry set values represents every extension field see thm quasilinear setting anisotropic form already recovered set values represents base field indeed following simple observation proposition prop let quasilinear quadratic forms pan qan particular pan qan need following easy consequence proposition corollary prop let quasilinear quadratic form let field extension exists subform qan proof obviously spanned space elements form basis proposition shows subform qan required property inseparable quadratic extensions make use basic results concerning behaviour quasilinear quadratic forms scalar extension inseparable quadratic extension ground field first note following statement direct consequence additivity property quasilinear quadratic forms lemma lem let quasilinear quadratic form let subsets following result elaborates upon lemma case inseparable quadratic extension directly analogous standard result characteristic theory see cor lemma let anisotropic quasilinear quadratic form let exist elements stephen scully forms let form see round particular set subspace subfield recall similarity factors form constitute subgroup moreover property characterises anisotropic quasipfister forms since forms characterised roundness norm form see let quasilinear quadratic form norm field denoted defined smallest subfield containing ratios elements evidently note also definition thus exists isometry unique anisotropic form qnor property qnor form called norm form note element represented aqan qnor indeed follows proposition light obvious inclusion aqan qnor dimension qnor important invariant known norm degree follows convenient work logarithm denote lndeg words lndeg dim qnor need following lemma lemma thm let quasilinear quadratic form lndeg lndeg similarity factors see another direct consequence proposition following statement concerning similarity factors lemma lem let quasilinear quadratic form let note quasilinear condition preceding lemma closed addition since subgroup follows subfield divisibility forms main results isotropy behaviour quasilinear quadratic forms function fields quadrics obtained studying extent certain forms divisible forms mind useful introduce related terminology see given quasilinear quadratic form define divisibility index denoted largest integer qan divisible quasipfister form higher divisibility indices defined divisibility indices higher anisotropic kernel forms respectively worth recording following easy calculation lemma lem let quasilinear quadratic form let purely transcendental field extension particular higher divisibility indices quasilinear quadratic form insensitive purely transcendental field extensions need observations regarding notion divisibility first analogous standard result cor regarding divisibility quadratic forms pfister forms fields characteristic lemma prop let quasilinear quadratic form anisotropic form divisible qan also divisible next well known char anisotropic quadratic form divisible anisotropic pfister form every field extension see thm cor quasilinear setting condition already checked base field compare proposition proposition prop cor let anisotropic quasilinear quadratic form anisotropic form divisible following consequence corollary let anisotropic quasilinear quadratic forms divisible pnor similar proof let proposition pnor divisible pnor since subfield containing see see divisible pnor lemma equivalent assertion since subspace equalities hold proposition preceding inclusion amounts hand proposition also shows assertion subform thus see divisible pnor pan proves desired assertion isotropy quasilinear quadratic forms field extensions record basic observations regarding isotropy behaviour quasilinear quadratic forms field extensions first concerns separable extensions say separable extension mean algebraic closure ring nilpotent elements lemma prop let anisotropic quasilinear quadratic form separable field extension anisotropic particular anisotropic forms remain anisotropic purely transcendental field extensions secondly need slightly subtle result regarding function fields affine hypersurfaces let system algebraically independent variables let irreducible polynomial let denote fraction field integral domain function field integral affine hypersurface given element let write multg largest integer added convention multg following lemma prop situation let quasilinear quadratic form let suppose anisotropic multg mod assertion lemma holds arbitrary quadratic forms arbitrary characteristic indeed may deduced easy consequence fundamental representation theorem cassels pfister thm however statement peculiar quasilinear case observed hoffmann cor one may use additivity quasilinear quadratic forms generalize theorem following stronger statement stephen scully theorem hoffmann cor let quasilinear quadratic form let system algebraically independent variables let main results quasilinear case final section prove several new results concerning isotropy behaviour quasilinear quadratic forms function fields totally singular quadrics main result theorem number interesting statements including quasilinear part theorem follow formally theorem essentially generalization thm proof follows latter closely however certain adjustments needed along way facilitate extra generality among need work certain generic subforms given anisotropic form therefore begin brief discussion around idea previous section denote arbitrary field characteristic throughout generic subforms following lemma well known though precise reference lemma let field let proper morphism irreducible generic fibre rational point every fibre regular locus rational point remark regarding fibre point scheme residue field particular rational point means proof assume generic fibre rational point let regular point reduced closure codimension discrete valuation ring statement follows otherwise regularity implies regular point integral closed subvariety statement therefore follows induction codimension using prove following existence statement needed proof theorem lemma let anisotropic quasilinear quadratic form dimension exist purely transcendental field extension subform purely transcendental extension remark remind reader denotes function field affine opposed projective hypersurface defined vanishing proof construct generic compare lem let underlying vector space let denote projective defined vanishing respectively let denote grassmannian hyperplanes consider closed subschemes respectively generic fibre canonical projection projective quadric defined vanishing subform since rational purely transcendental extension claim satisfies conditions suffices show purely transcendental extension purely transcendental extension see note however also canonically function field generic fibre projection latter fibre nothing else projective space hyperplanes containing canonical rational point spec follows note since quasilinear quadratic form representing anisotropic proposition implies dimension reasons enough check anisotropic vanishing locus nothing else generic fibre canonical projection second incidence scheme since latter projection proper projective fibres regular points posses rational point anisotropic exists hyperplane represent generic fibre admits rational point lemma words anisotropic claimed remark situation lemma let shown totaro thm assertion holds see also cor use fact follows subform condition isotropy quasilinear quadratic forms function field quadric following theorem main result section remark replacing purely transcendental extension qualification may removed see remark theorem let anisotropic quasilinear quadratic forms dimension isotropic replacing purely transcendental extension exists anisotropic quasilinear quadratic form dim proof multiplying appropriate scalars necessary may assume forms represent lemma find purely transcendental field extension subform affine function field purely transcendental extension fix pair remainder proof let choose elements set system algebraically independent variables field may identified fraction field integral domain see fixing identification henceforth write image given polynomial canonical projection polynomial also write multiplicity largest integer note simplify notation let hypothesis proof theorem begins following lemma lemma assume situation exists subform elements stephen scully anisotropic form defined anisotropic proof existence subform satisfying ensured corollary since purely transcendental extension also satisfies view lemma purely transcendental extension field purely transcendental extension hence another application lemma see therefore shows since lemma implies multiplying squares necessary may arrange thus exists sequence holds among sequences let choose one minimal polynomial denotes degree viewed polynomial variable field claim choice also holds suppose sake contradiction case form isotropic exist polynomials iii least one indeed andp iii amount stated isotropy form arranged field closed addition choice since anisotropic lemma implies note indeed case iii would imply isotropic since latter form similar subform anisotropic form furthermore since since closed multiplication squares taking generalized representation theorem quasilinear quadratic forms account theorem see fact iii exists integer among let choose one integer maximal since proposition together implies forms words sequence elements satisfying condition since choice imply contradicts original choice conclude supposition incorrect lemma proved let fix subform elements satisfying four conditions lemma since purely transcendental extension theorem follows following precise lemma lemma let situation proof proposition statement equivalent assertion sides spaces suffices show side contains set generators side since generated therefore sufficient show bfi fact suffices show case indeed since space construction contains bfi order check statement holds first need following preliminary calculation sublemma assume situation let exist bfi least one element proof simplicity notation let consider field may identified function field affine quadric defined vanishing hbi since since view proposition follows particular purely transcendental extension see thus purely transcendental extension lemma choice follows proposition means since square since subset see lemma follows write every element evidently ratio element square therefore find elements let min defined beginning proof since anisotropic lemma shows hence even particular let replacing pair pair alter arrive situation pairs least one entry wanted returning proof lemma let let sublemma bfi stephen scully pairs least one entry claim equivalently see let first clear denominators preceding equation obtain equality bfi polynomial ring let min claim equivalent assertion suppose case let minimal integer complementary reducing sides modulo see mod since pair least one entry follows equivalently contradicts supposition claim therefore valid reducing modulo dividing obtain bfi shows bfi needed per discussion theorem proved remark already mentioned replacing purely transcendental extension may removed statement theorem indeed choose pair used throughout proof see remark unclear author whether qualification really needed case statement proved sufficient basic applications applications provide concrete applications theorem problem understanding splitting behaviour quasilinear quadratic forms scalar extension function field quadric particular prove quasilinear case theorem reader note case form plays similar role played upper motive quadric proof characteristic case given particular importance remember dimension dim quasilinear setting opposed familiar dim indeed quasilinear quadratic form dim dim see divisibility indices introduced also key role play effect study indices replaces implicit use steenrod operations chow groups characteristic setting proceeding make general remark remark let anisotropic quadratic forms dimension quasilinear quasilinear associated quadric generically smooth amounts assertion function field separable extension see view lemma therefore case result studying isotropy behaviour quasilinear form scalar extension field case interest also quasilinear first interesting application theorem theorem one obtains short proof quasilinear part theorem proving theorem first make following quick observation lemma let anisotropic quasilinear quadratic forms let purely transcendental field extension similar subform similar subform proof finite perfect necessarily split since statement trivial case may assume infinite furthermore problem easily reduced case finite transcendence degree using proposition therefore also assume system algebraically independent variables similar subform evidently find polynomial since infinite exist scalars letting claim proposition suffices show since certainly generalized representation theorem quasilinear forms theorem follows claim follows performing specialization well defined polynomial ring remark fact quasilinearity hypothesis necessary one may show statement holds arbitrary pair anisotropic quadratic forms field characteristic using induction transcendence degree general representation theorems pfister thm thm refrain going details theorem let anisotropic quasilinear quadratic forms dimension isotropic similar subform proof lemma sufficient show replacing purely transcendental extension theorem may therefore assume exists anisotropic form dimension since anisotropic proposition implies proves theorem consequence theorem get elegant explanation quasilinear part theorem result originally proved totaro using basic machinery chow groups see loc thm elementary proof rather different one presented later given corollary totaro let anisotropic quasilinear quadratic forms dimension isotropic dim dim proof theorem similar subform particular dim dim dim dim rearranging inequality obtain desired assertion next applications make use fact form appearing statement theorem dimension least begin main result result used loc determine possible values knebusch splitting pattern quasilinear forms includes quasilinear part theorem immediate consequence corollary thm let anisotropic quasilinear quadratic form dimension words divisible form dimension stephen scully proof lemma statement insensitive replacing purely transcendental extension applying theorem case therefore assume anisotropic form dimension per proof theorem however similar subform dimension reasons therefore conclude corollary implies divisible form since anisotropic similar subform see particular dim dim completes proof given corollary generalize follows theorem let anisotropic quasilinear quadratic forms dimension isotropic dim dim words divisible form dimension given hypotheses proof lemma statement insensitive replacing purely transcendental extension thus theorem may assume exists anisotropic form dimension let proof corollary similar subform theorem follow exactly arguments show two forms dimension given hypotheses suppose sake contradiction case dim dim indeed since definition divisible anisotropic form dimension lemma implies true thus dim dim divisible claim follows particular since dim dim dim dim dim corollary asserts together previous inequality gives dim dim contradicts original hypothesis thus completes proof theorem taking dimensions obtain quasilinear part theorem light remark enough treat case quasilinear corollary let anisotropic quasilinear quadratic forms dimension max dim dim particular setting dim max dim dim proof first statement follows immediately theorem second follows first since definition divisor dim dim remark situation corollary inequality need equality example generic dimension positive integer see lem theorem interestingly know analogue integer characteristic theory fact proof theorem shows corollary may refined follows corollary let anisotropic quasilinear quadratic forms dimension max dim dim remark refinement happens example indeed case odd corollary corollary deduced statement theorem identifying certain situation subform similar another interesting problem find circumstances similar proof next proposition shows one situation happens close attaining maximal possible value dim unfortunately formulating means general terms necessitates certain degree technicality hope however subsequent discussion help illuminate concrete meaning observation stating proposition recall see anisotropic quasilinear form lndeg denotes integer dim pnor since similar subform pnor dim proposition let anisotropic quasilinear quadratic forms dimension let unique integer mod dim possibly replacing purely transcendental extension contains subform dim lndeg divisible form dimension proof theorem may assume exists anisotropic form dimension claim given hypotheses similar exactly proof theorem certainly similar subform therefore suffices show dim dim suppose sake contradiction case dim indeed definition divisible form dimension lemma therefore true thus dim divisible since dim inequality dim follows definition since subform yields dim dim dim contradicts hypothesis therefore conclude similar corollary follows divisible form latter form dimension lemma thus choosing see required properties proposition immediately gives following result theorem let anisotropic quasilinear quadratic forms dimension let unique integer mod either dim divisible stephen scully remark corollary thus larger interesting restrictions theorem become example let anisotropic quasilinear quadratic forms dimension dim maximal possible value situation analogous characteristic one form becomes hyperbolic function field see cor dim lndeg dim dim divisible power least large hence dim see indeed situation necessarily case theorem since dim claim follows immediately fact case one show divisible form pnor dimension see thm theorem may therefore viewed generalization result giving necessary conditions order close maximal value index close maximal value analogous situation characteristic one form becomes almost hyperbolic function field another quadric setting known general results spirit theorem example illuminate theorem working concrete example namely neighbour dim similar subform anisotropic form example could form case lndeg see cor details theorem therefore implies anisotropic quasilinear quadratic form dimension either dim divisible unique integer mod fields characteristic statement known hold special case dim indeed classical result due arason pfister asserts quadratic form becomes hyperbolic function field pfister quadric necessarily divisible corresponding pfister form see cor however seems known generalization result similar one given quasilinear forms acknowledgements author gratefully acknowledges support pims postdoctoral fellowship nserc discovery grant preparation article references elman karpenko merkurjev algebraic geometric theory quadratic forms ams colloquium publications american mathematical society grothendieck avec collaboration jean globale quelques classes morphismes publications grothendieck avec collaboration jean locale des des morphismes des partie publications fitzgerald function fields quadratic forms math haution first steenrod square chow groups amer haution detection regular schemes degree two algebr geom hoffmann laghribi quadratic forms pfister neighbors characteristic trans amer math hoffmann laghribi isotropy quadratic forms function field quadric characteristic algebra hoffmann isotropy quadratic forms function field quadric math hoffmann diagonal forms degree characteristic algebraic arithmetic theory quadratic forms contemp amer math izhboldin virtual pfister neighbors first witt index introduction nikita karpenko geometric methods algebraic theory quadratic forms lecture notes springer bruno kahn formes quadratiques sur corps cours france paris karpenko first witt index quadratic forms invent karpenko canonical dimension proceedings international congress mathematicians volume hindustan book agency new delhi karpenko upper motives algebraic groups incompressibility varieties reine angew nikita karpenko alexander merkurjev essential dimension quadrics invent knebusch generic splitting quadratic forms proc london math soc knebusch generic splitting quadratic forms proc london math soc laghribi totally singular quadratic forms algebraic arithmetic theory quadratic forms contemp amer math rost new results chow groups quadrics preprint scully rational maps quasilinear hypersurfaces compos scully hoffmann conjecture totally singular forms prime degree appear algebra number theory pages scully splitting quasilinear reine angew totaro birational geometry quadrics characteristic algebraic vishik integral motives quadrics mpim preprint vishik direct summands motives quadrics preprint https stephen scully vishik motives quadrics applications theory quadratic forms geometric methods algebraic theory quadratic forms lecture notes springer vishik excellent connections motives quadrics ann sci norm department mathematical statistical sciences university alberta edmonton canada address stephenjscully
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comparison policies participation storage frequency regulation markets bolun student member ieee yury dvorkin student member ieee daniel kirschen fellow ieee member ieee watson member ieee feb emails xubolun dvorkin kirschen casilv jwatson energy storage systems better ramping characteristics traditional generators participation frequency regulation facilitate balancing load generation however sustain output indefinitely system operators therefore implemented new frequency regulation policies take advantage fast ramps energy storage systems deliver alleviating problems associated limited energy capacity paper contrasts several policies directly affect participation energy storage systems frequency regulation compares revenues owners systems might achieve policy index storage power system economics power system frequency control ntroduction frequency regulation service involves injection withdrawal active power power grid maintain system frequency united states frequency regulation equivalent secondary frequency control primary frequency control commonly known frequency response common mean unit provide frequency regulation following system operator automatic generation control agc signal computes area control error ace frequency deviations interchange power imbalances appropriate response agc signal requires dispatching flexibility within assigned regulation capacity traditionally majority frequency regulation capability provided specially equipped generators technologies evolve new types flexibility resources emerge battery flywheel energy storage energy storage systems significantly faster ramping capabilities compared conventional generators participation frequency regulation reduce need procure regulation capacity create incentive fast responding units participate frequency regulation federal energy regulatory commission ferc enacted ferc order requires system operators add performance payment accuracy adjustment capacity payment typically used markets ancillary services scheme implemented independent system operators iso regional transmission organizations rto new market rule fast ramping regulation resources receive higher regulation payments paper presented ieee pes general meeting performance usually accurate case study conducted res americas already shown mwh battery storage achieved high performance following regulation signal provided pjm utilize fast responsive capability energy storage systems ess beyond traditional agc framework system operators pjm introduced fast regulation units participating fast regulation follow regulation signal changes much faster traditional agc signal receive extra payments system operator also benefits improved regulation accuracy review california energy storage alliance cesa shows combination fast regulation reduced procurement requirement pjm regulation market however high investment cost ess far limited installed energy storage capacity energy lost charging discharging energy imbalance regulation signal make difficult ess operators provide regulation services seamlessly relatively long time period several hours system operators elected lower barrier energy storage dispatching regulation taking account state charge soc cycling efficiency ess paper compares market policies affect energy storage frequency regulation market covers isos rtos organized wholesale electricity markets including pjm interconnection pjm new york independent system operator nyiso midcontinent independent system operator miso iso new england isone california independent system operator caiso study results fast regulation storage energy compensation discussed section section iii section respectively section analyzes regulation payment different markets section summarizes findings erformance iso rtos study complied ferc order implemented regulation markets except nyiso miso pay regulation capacity regulation markets dam typical regulation market involves following terms table market terms price regulation mileage performance factor capacity mileage pjm capability price performance price mileage performance score nyiso capacity price movement price movement performance index miso capacity price mileage price mileage performance accuracy capacity price service price movement performance score caiso capacity price mileage price mileage accuracy percentage applies caiso regulation market nyiso also applies miso unit performance unit receives full credit performance test interval performance factor used adjust capacity payment thus becomes regulation capacity maximum minimum regulation power unit offers given period regulation capacity usually symmetrical unit provides positive negative capacity exception caiso separates regulation components capacity price represents option price reserve capacity provide regulation future confused exercise price energy required follow regulation signal regulation mileage sum absolute values regulation control signal movements mileage regulation resource output time period signal steps regulation capacity pmax unit mileage representing mileage regulation capacity unit mileage price performance factor score indicates unit performance following regulation signal model performance factor unit accuracy signal following integration absolute errors pjm uses complex measure performance involves delay correlation accuracy table shows names used quantities market market participants offer capacity price bid mileage price bid system operator credits providers regulation using market capacity clearing price mileage clearing price performance factor unit procurement period unit must reach minimum performance score qualified receiving credits maintain minimum performance certain period eligible bid regulation market future generalize method awarding regulation credit follows pjm uses mileage ratio credit calculations mileage ratio ratio mileage assigned regulation signal mileage traditional regulation signal mrega see section formula calculating credit becomes mrega iii fast egulation fast regulation signal generated applying highpass filter agc signal signals fast ramping rate designed therefore ideal ess interested providing frequency regulation fast regulations implemented pjm miso provides alternative approach utilizing fast responsive units regulation signal energy capacity requirement refers minimum energy capacity needed follow perfectly regulation signal neglecting efficiency soc constraints ess energy requirement signal hour difference maximum minimum soc operating hour regulation signal hour step size steps per hour hourly energy requirement therefore min max energy balance regulation signal initial soc level allows storage minimum energy capacity follow signal perfectly ignoring efficiency soc constraints perfectly balanced signal allows soc level energy storage return initial level end operating period since assumes initial soc zero simulates ess negative capacity energy balance signal min pjm regd pjm provides two regulation signals regulation market rega filtered ace designed traditional regulation units regd filtered ace designed fast regulation units regd signal controlled within minutes units choose freely signal follow follow one signal time analysis regulation signals miso mwh pjm regd isone enc isone ent pjm rega isone base regulation energy hourly requirement capacity miso reserve market deployment interval ess called stored energy resources ser optimized ess based initial condition taking things one step miso proposing new regulation reserve deployment scheme called idea scheme deploy fast ramping units first agc dispatch creating new fast ramping deployment group miso expects improve utilization fast ramping units torage nergy ffset ompensation soc pjm regd isone enc isone ent pjm rega isone base regulation energy hourly balance fig comparison pjm regulation signals june may shows mileage rega per hour mileage regd around per hour suggesting regd ramps much faster market settlements mileage ratio rega therefore mileage ratio regd around section provides detailed comparison regulation payments box plot fig summarizes distribution hourly energy requirement shows cases pjm regd energy requirement less mwh approximately minutes full fig shows regd average energy balance extreme cases reach includes ess alternative technology regulating resources atrr provides two fast regulation signals continuous enc trinary ent conventional regulation signal resource change month signal follows enc identical pjm regd signal ent specifically designed atrrs dispatch either zero full charge power full discharge power since trinary dispatched resource assumed step response hence equal participation factors ent algorithm dispatches entire set trinary dispatched atrrs together fig shows energy requirement balance analysis using duration simulated signal suggest signals behave similar fashion pjm regd rega mileage difference three simulated signals significant average mileages enc ent conventional signal respectively due less perfect efficiency fact agc signal mean soc ess maintained traditional regulation framework fast frequency regulation alleviates effect signal energy imbalance isos decided directly compensate energy offset storage units regulation market dispatching mechanisms include regulation energy management rem caiso dispatching rtd nysio caiso rem caiso treats ess resources ngr set resources also includes responsive demands ngr units two options participating regulation market traditional rem units subjected requirements traditional regulation units must meet continuous energy requirement provide regulation capacity nonrem ngr unit must able deliver least mwh continuous energy requirement units minutes units participate caiso regulation market subscription rem changed monthly caiso compensates energy offset units participating rem ngr unit submits preferred soc caiso maintain dispatching energy real time market rtm next real time dispatch interval balancing energy dispatched units together agc signal energy management system ems ems also takes account unit efficiency dispatch rem energy compensation guaranteed normal operations emergency cases soc restored later caiso regulation market settlement regulation energy charged discharged storage dispatched rem energy settled rtm locational marginal price lmp addition capacity settlement mileage settlement since rem guarantees maintain soc ngrs consequence energy settlement units must pay energy losses lmp settlement method currently unique caiso pjm regd pjm rega isone ent miso nyiso caiso payment fig dam regulation price comparison outliers omitted nyiso rtd nyiso calls ess limited energy storage resources lesr lesr bid regulation dam rtm receive mileage payments rtm settlement regulation market lesr units evaluated scheduled hourly basis without considering potential energy limitations regulation market rtd manages energy level lesr maintain regulation capability extent possible charging discharging lesr necessary rtd assigns regulation base points lesr every minutes based solely soc level example interval regulation capacity lesr provide regulation reserve base point soc within centered soc soc provide regulation reduced capacity positive charging base point soc provide regulation reduced capacity negative charging base point thus soc lesr converges caiso rtd soc management suspended emergencies called corrective action mode lesr providing regulation discharge zero order make loss generation soc level restored terminates egulation arket ncome market results section compares regulation payments different regulation markets using public regulation pricing data markets assume performance factor ideal use eqs calculate payments box plot fig shows distribution calculated regulation payments dam pjm payments two regulation products available pjm based signals market clearing prices june may hourly regulation mileage signal calculated using caiso caiso publishes historical market data caiso open access information system oasis besides pricing data caiso also publishes hourly mileage data system mileage multipliers agc signal consistent calculations markets assume unit offers regulation regulation capacity calculate payments separately using market prices june may add two results together average calculated price price miso miso publishes ancillary service pricing data market report website use ancillary service market prices june may market mileage price publishes ancillary service pricing data market data information website use final hourly clearing price data june may started mileage bidding march clearing price april may contains capacity price service mileage price two months use mileage simulated ent signal described section payment two signals calculated since mileages close ent nyiso nyiso publishes ancillary service pricing data pricing data website use dam regulation clearing prices june may market mileage price market caiso miso nyiso rtm dam operating reserves rtm procure additional reserves according system updated system conditions three markets mileage payments however mileage clearing prices close zero thus little effect final price caiso caiso determines rtm regulation capacity scheduling process weighted average rtm price rtm prices respectively miso miso rtm cleared every five minutes produce price dispatch levels regulation annual average hourly rtm regulation prices general higher dam prices nyiso nyiso calculates rtm requirement rtd interval hour every minutes average dam regulation price average rtm regulation price onclusions table summarizes findings study column ancillary service requirements list minimum energy capacity required provide regulation reserve requirement specified caiso pjm derived simulations described section iii able find similar information miso nyiso last three columns show table comparison regulation market policies energy storage fast responsive units requirements cap mwh offset control fast reg schemes abbr reg cap reg cost avg ngr caiso rem else rem pjm regd rega regd else atrr ser lesr miso nyiso rtd average required regulation capacity regulation cost average generation capacity operator zone comparison shows pjm operates efficient regulation market terms ratio regulation capacity generation cost regulation similar isos relative size interconnection particular compare ancillary market cost pjm caiso historical cost regulation pjm per mwh load cost caiso hand total per mwh ancillary service cost much higher pjm caiso possible explanation difference caiso performs energy ancillary services pjm coupled process complicated categories ancillary services latter approach less efficient caiso results higher ancillary service cost example sometimes pjm regd capacity meaning capacity ratio regd high since regd logic maintain energy high regd capacity ratio make controller ace correlation extreme cases comparison different markets shows regulation markets becoming suitable profitable ess new policy makes energy storage competitive new regulation policies lowered barrier ess provide regulation longer periods perspective system operator participation energy storage appropriate market designs makes regulation efficient however providing regulation signal challenging must well coordinated ancillary service products energy balancing signal perturb system acknowledgment authors thank gyuk colleagues doe energy storage program funding research sandia national laboratories laboratory managed operated sandia corporation wholly owned subsidiary lockheed martin corporation department energy national nuclear security administration contract eferences dvorkin kirschen assessing flexibility requirements power systems iet generation transmission distribution vol makarov nguyen assessing value regulation resources based time response characteristics pacific northwest national laboratory tech online available http ferc order frequency regulation compensation organized wholesale power markets case study pjm frequency online available http edgette pay performance regulation year design changes tech online available http pandzic wang qiu dvorkin kirschen nearoptimal method siting sizing distributed storage transmission network ieee transactions power systems vol sept pjm manual balancing operations pjm online available http pjm manual energy ancillary services market operations pjm online available http pjm manual operating agreement accounting pjm online available http nyiso accounting billing manual online available http miso business practices manual market settlements online available https chen leonard keyser gardner development regulating reserve compensation miso energy ancillary service market power systems ieee transactions vol iso new england market rule online available http hinman pay performance regulation online available http brief history regulation signals pjm online available http description energy neutral agc dispatch online available http simulated automatic generator control agc setpoint online available http miso business practices manual energy operating reserve markets online available https agc enhancement fast ramping resources online available https california iso resource ngr regulation energy management rem online available http business requirements specification caiso online available http nyiso ancillary services manual online available http hickey limited energy storage resource market integration update online available http miso market online available https pricing online available http nyiso pricing online available http ellison tesfatsion loose byrne project report survey operating reserve markets electric energy regions sandia national laboratories tech caiso annual report market issues performance online available http miso annual market assessment report online available https state market report new york iso electricity markets online available http business practice manual market operations caiso online available https pjm state market monitoring analytics online available http iso new england operating procedure online available http iso new englands internal market monitor annual markets report online available http state market report miso electricity markets online available https
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planning using approximate dynamic programming enhance community resilience management saeed yugandhar bruce ellingwooda edwin chongb hussam mahmouda dept mar dept civil environmental engineering colorado state university fort collins electrical computer engineering colorado state university fort collins abstract lack comprehensive approach significant importance must garnered immediate attention algorithms need solve largescale optimization problems pose computational challenges complexity optimization problems increases various sources uncertainty considered research introduces sequential discrete optimization approach framework recovery management proposed mathematical approach leverages approximate dynamic programming along heuristics determination recovery actions methodology overcomes curse dimensionality manages infrastructure systems following disasters also provide computational results suggest methodology incorporates recovery policies responsible public private entities within community also substantially enhances performance underlying strategies limited resources methodology implemented efficiently identify recovery decisions following severe earthquake based multiple objectives modeled electrical power network testbed community coarsely modeled gilroy california united states proposed optimization method supports community within chaotic posthazard circumstances keywords community resilience electrical power network combinatorial optimization rollout algorithm approximate dynamic programming introduction modern era functionality infrastructure systems significant importance providing continuous services communities supporting public health safety natural anthropogenic hazards pose significant challenges infrastructure systems cause undesirable system malfunctions consequences past experiences show malfunctions always inevitable despite design strategies like corresponding author email addresses saeed nozhati yugandhar sarkale bruce ellingwood edwin chong hussam mahmoud contributed equally listed alphabetically preprint submitted reliability engineering system safety increasing system redundancy reliability therefore sequential rational framework enable malfunctioned systems restored timely manner hazards stressors chaotic circumstances time limitations budget resource constraints complexities community recovery process twinned catastrophe highlight necessity comprehensive riskinformed framework recovery management comprehensive framework must take account indirect delayed consequences decisions also called property decisions thus requires foresight planning comprehensive system must also able handle scheduling problems encompass large combinatorial decision spaces make rational plans communitymarch level developing efficient computational methodologies sequential problems always subject context civil engineering several studies utilized framework dynamic programming management bridges pavement maintenance typical methodological formulations employ principles dynamic programming utilize pairs study show powerful relatively unexplored methodological framework formulating large infrastructure problems described section formulation require explicit model therefore shielded common problem state explosion methodologies employed sequential methodology presented manages infrastructure also considers interconnectedness cascading effects entire recovery process addressed past studies dynamic programming formulations frequently suffer curse dimensionality problem aggravated deal large combinatorial decision spaces characteristic community recovery therefore using approximation techniques conjunction dynamic programming formalism essential several approximation techniques available literature use promising class approximation techniques called rollout algorithms show rollout algorithms blend naturally formulation together form robust tool overcome limitations faced application dynamic programming techniques massive problems proposed approach able handle curse dimensionality analysis management infrastructure systems data sources proposed methodology also able consider improve current recovery policies responsible public private entities within community among infrastructure systems electrical power networks epns particularly critical inasmuch functionality networks critical facilities depend epn functionality management hence method illustrated application recovery management modeled epn gilroy california following severe earthquake illustrative example shows posed approach implemented efficiently identify recovery decisions computed decisions restored epn gilroy timely manner residential buildings well main food retailers example critical facilities need electricity support public health aftermath hazards remainder study structured follows section introduce background system resilience system modeling used study section introduce case study used paper section describe earthquake modeling fragility restoration assessments section provide mathematical formulation optimization problem section demonstrate compatibility rollout algorithm string action formulation multiple simulations section present brief conclusion research system resilience term resilience defined variety ways generally speaking resilience defined ability prepare adapt changing conditions withstand recover rapidly disruptions hence resilience community system usually delineated measure community functionality shown vertical axis four attributes robustness rapidity redundancy resourcefulness illustrates concept functionality defined ability system support planned mission example providing electricity people facilities understanding interdependencies among components system essential quantify system functionality resilience interdependencies produce cascading failures cascade may triggered malfunction single components contribute recovery rate difficulty entire recovery process system different factors affect recovery rate system among modification disruptive events mitigations different recovery policies actions nature disruption prominent also highlights different sources uncertainty associated community functionality assessment remarkable impacts different stages prior event end recovery process therefore employed model assess recovery figure map gilroy population defined grids figure schematic representation resilience concept adopted resolution sufficient study methodology hazard events gilroy contains food retailers six main retailers employees provide main food requirements gilroy inhabitants shown summarized table cess ought able consider impacts influencing parameters study dependency networks modeled adjacency matrix xij xij indicates magnitude dependency components general form adjacency matrix timedependent stochastic matrix capture uncertainties dependencies probable timedependent variations according literature resilience index system defined following equation tlc table main food retailers gilroy food retailer number employees walmart costco target pueblo food nob hill foods safeway order assign probabilities shopping activity urban grid gravity model used gravity model identifies shopping location probabilistically given location residences probabilities assigned proportional food retailers capacities inversely corresponding retailers distances centers urban grids consequently distant small locations less likely selected close large locations center urban grid food retailer chosen according following distribution ebtcr shaded area underneath functionality function system time tlc control time system time occurrence event shown fig description case study case study paper community gilroy california usa used example illustrate proposed approach gilroy located approximately kilometers south city san jose population time census see study area divided grids define properties community encompasses area gilroy characteristics gilroy covered study adopted model capacity food retailer determined table negative constant tcr travel time urban grid food retailer googles distance matrix api called within using ggmap package provide distances travel time assumed transportation mode driving represents electrical power network figure map shear velocity gilroy area loma prieta earthquake spatial estimation amplitudes earthquakes essential element risk assessment typically characterized prediction equations gmpes gmpes require several parameters earthquake magnitude fault properties soil conditions average velocity top soil epicentral distances compute seismic intensity measure point modern gmpes typically take form figure gilroy main food retailers reflect within event uncertainty respectively study gmpe proposed ambrahamson used ground motion similar loma prieta earthquake one devastating hazards gilroy experienced epicenter approximately gilroy downtown saf projection simulated show map ground motion field peak ground acceleration pga respectively figure modeled electrical power network gilroy epn components located within defined boundary llagas power substation main source power defined boundary supplied transmission line distribution line components positioned modeled substation urban grids centers food retailers water network facilities study modelled epn components fragility function restoration relations intensities earthquake damage relationships pivotal elements quantification structures infrastructure systems functionality loss estimation risk analysis community fragility curves describe probability experiencing exceeding particular level damage hazard damage assessment earthquake simulation seismicity gilroy region california mainly controlled san andreas fault saf caused numerous destructive earthquakes like optimization epn recovery gilroy problem description introduction earthquake event occurs epn component experiences one damage states shown table let total number damaged components note also study assumption justified availability limited resources planner large number components damaged aftermath severe hazard decision maker planner task assigning units resources damaged components decision maker heuristic expert policy basis make decisions maximize objective precise nature objective planner vary described details section particularly decision epoch decision maker resource planner deploys unit resources damaged components unit resources assigned distinct damaged component study assume outcome decisions fully predictable consider separate article address relaxation assumption outcome decisions exhibits uncertainty modified methods deal general problem form part separate paper planner must option reassigning resources new locations based heuristics objectives possible preempt assignment unit resources damaged locations however choose solve general problem preemptive assignment locations special case problem preemptive assignment problem richer decision problem case sense process optimizing decision actions complex task since size decision space bigger improve upon solutions offered heuristics planner formulating optimization problem dynamic program solving using rollout approach figure simulation median peak ground acceleration field function hazard intensity customary model component fragilities lognormal distributions conditional probability exceeding particular damage state conditioned particular level intensity measure defined standard normal distribution mean standard deviation fragility curves obtained based damage evaluation data empirical curves structural modeling analytical curves expert opinions heuristics curves present study seismic fragility curves included study xie used illustration order restore network number available resource units generic single number including equipment replacement components repair crews considered assignment damaged components damaged component assumed require one unit resources restoration times based level damage used study presented table based optimization problem section provide formal representation optimization problem lets say decision maker starts making decisions assigning repair locations different units resources number decisions table restoration times based level damage component electric transmission line component distribution line component damage states undamaged minor moderate extensive complete made becomes sequential application repair actions damaged components assignment units resources becomes trivial problem setting unit resources simply assigned one one order damaged components consequently strict optimal assignment achieved trivial case size trivial assignment problem reduces one every new decision epoch remaining damaged components repaired additional units resources retire deploying one unit resource location decrease repair time associated damaged component henceforth focus assignment problem let variable denote decision epoch let set damaged components repair action performed let tend denote decision epoch note tend components repaired let xtend represent set string actions say repair action completed least damaged components repaired let powerset let objective compute optimal solution given arg min explain precise meaning restoration electricity people detail sum objective aim find set string actions minimizes number days needed restore electricity certain fraction total population gilroy objective define mapping terms number people electricity per unit time objective maximize mapping string repair actions let variable denote total time elapsed completion repair action time elapsed start completion repair action let total number people electricity repair action complete let dtend end ktend interested optimal solution given let set repaired components repair action completed note decision making problem moves trivial assignment problem previously discussed wish calculate string repair actions optimizes objective functions deal two objective functions study denoted mapping arg max note objective function case mimics resilience index interpreted terms eqn integral replaced sum discrete decision epochs replaced product tlc analogous ktend integral limits also changed represent discrete decision epochs also note decision maker may apply repair actions sequentially different decision epochs maximize minimize cumulative objective function string actions represented outcome sequential decision making process particularly relevant context dynamic programming objective let variable represent population gilroy represent constant threshold let string repair actions results restoration electricity number people number damaged component ith decision epoch let represent time required restore electricity number people result repair actions let formally approximate dynamic programming possible use dynamic programming formalism represent optimization problem accomplish starting first principles ref let focus attention objective extension methodology objective straightforward need adapt notation used objective methodology presented maximization problem replaces minimization problem recall interested optimal solution given calculated following manner first calculate follows arg min called curse dimensionality consider objective wish calculate damaged components unit resources table size practice replaced approximation denoted approximation constructed aid heuristic several ways utilize heuristic approximate approximation techniques fall category approximate dynamic programming technique rollout one heuristic approximation process let say heuristic available used approximate minimization let denote corresponding approximate optimal value rollout yields suboptimal solution replacing eqn function defined min next calculate arg min arg min rollout results significant improvement applying base heuristic solve approximate dynamic programming problem function defined min rollout algorithm problem discrete deterministic case base policy base heuristic dynamic programming sense see let call base heuristic defined several ways similarly calculate solution follows arg min function defined min current recovery policy regionally responsible public private entities functions called optimal functions defined following recursion min importance analyses prioritize importance components based considered importance factors iii greedy algorithm computes greedy heuristic boundary condition given random policy without note standard notation used represent functions dynamic programming literature approach discussed calculate optimal solutions dynamic programming formulation however except special problems formulation solved exactly calculating storing optimal functions numerically intensive particularly problem let denoted storage requires table size dynamic programming literature empirical policy base heuristic based maximum node link betweenness shortest path example used studies restriction define base heuristic use proposed formulation wiser base heuristic potentially enhance computed rollout policy two different base heuristics considered study first base heuristic random heuristic denoted idea behind consideration heuristic actuality cases thoughtout strategy computation computationally extensive show rollout formulation accept random base decision maker improve significantly second base heuristic called smart heuristic since computed based importance components expert judgment connoted important components based contribution community repaired first type base heuristic determination similar spirit items proposed formulation improve bases rollout policy obtained repeated use base heuristic approach used authors ref repeatedly used base heuristic improve solution term rollout first coined tesauro ref reference creating computer programs play backgammon approach similar rollout also shown let generate string actions let denote starting generated applying used evaluate due algorithm equal terminal reward therefore use heuristic find approximate solution problem approximation algorithm termed rollout due structure similar rollout algorithms algorithm generates string actions follows arg min repair action next two steps minimizing given repair actions next two steps viewed lookahead approach note similarity approach dynamic programming approach described except difficulty estimating values exactly also note twostep lookahead approach computationally intensive approach principle possible extend arbitrarily large step size however increases computational complexity algorithm increases exponentially particularly algorithm finds exact optimal solution exhaustively searching possible repair string combinations computational complexity also note provides tighter upper bound optimal objective value compared bound obtained original heuristic approach decision epoch explanation objective except maximize step given repair action results discussion show simulation results different cases case assume people electricity household units electricity household units people electricity components epn network serving household units either undamaged repaired functional recall city divided different grids according population heat maps different components epn network serving grids depicted essence entire population living particular grid electricity epn components serving grid functional therefore probability critical facility like food retailer urban grid electricity algorithm sequentially improving respect outperforms algorithm described lookahead approach objective every decision epoch repair action step optimized minimizing given repair action step possible generalize approach incorporate lookahead particularly let say optimize number epn component source destination lth epn component within defined boundary case along household units incorporate food retailers analysis say people figure electrical power network gilroy benefit electric utility epn components serving grids functional food retailer functional food retailer functional electric utility sense epn components serving retailers functional mapping number people access particular food retailer done grid level follows gravity model explained simulation results provided henceforth number units resources available planner fixed consider base heuristics simulation study base heuristic random smart base heuristic figure electrical power network tree form base heuristic also note stream branch epn network successive node dependent upon prior node electricity furthermore root components different branches dependent leaf component branches one branch dependent another one exploit serial nature epn representing epn network tree structure shown non trivial assignment problem number damaged components root node epn tree greater would unwise assign unit resources first decision epoch fringe nodes epn tree since get benefit repair damaged components root node soon number damaged components root node epn tree less explore assignment problem locations heuristic increase pool possible candidate locations assignment units resources must considered damaged locations next level epn tree number damaged locations current level epn tree less even considering next level epn tree case repair action optimization epn household units since consider epn household units section possible come heuristics reduce size search space rollout algorithm note heuristics base heuristic formulated reduce size present two heuristics motivate giving example let say access decision maker heuristic preordained method assigning units resources damaged locations let name components epn serial numbers shown partially per base heuristic decision maker component number damaged decision maker assign one unit resources component first schedule repair component contingent availability resources note base heuristic assumed illustration method method incorporate figure electrical power network recovery due base heuristic figure comparison base heuristic rollout algorithm heuristic rollout algorithm heuristic multiple scenarios objective number damaged components less take one step account damaged locations two levels current level repeat pool candidate locations greater equal ton levels epn tree exhausted heuristic note might possible ignore nodes level altogether assign units resources promising nodes current explored level epn tree achieved heuristic specifically number damaged components current level epn tree less algorithm searches node next level least number damaged components add damage components pool damaged components check total number damaged components explored levels less process repeated pool damaged components greater equal levels epn tree exhausted essentially consider set damaged components decision epoch subset denoted heuristic heuristic therefore plot epn recovery multiple scenarios used decision making objective pursued decision maker objective black line shows mean recovery trajectories red lines show standard deviation henceforth various plots instead plotting recovery trajectories scenarios compare mean different trajectories shows figure histogram base rollout rollout heuristic multiple scenarios performance algorithm respect simulation results demonstrate significant improvement algorithm used case case another result performance shown heuristic respect heuristic even though heuristic considers candidate locations case assignment problem decision epoch performance improvement shown minimal note simulate multiple damage scenarios following earthquake models discussed previously average total number damaged components scenario show performance base heuristic used rollout algorithm figure shows works considered simultaneously analysis due fact heuristic would simultaneously balance pruning damaged locations serving food retailers household units way achieving case heuristic household units considered whenever consider complex objective functions interaction networks difficult prune action space physically meaningful way applying heuristics case heuristic incorporate gravity model explained heuristic must also consider network topology actual physical position important epn components within network previously seen methodology works well even select small subset dtn construct avoid huge computational cost moreover degradation performance choosing dtn minimal despite methodology leverages use one step lookahead property consistent sequential improvement algorithm overcome degradation performance result choosing justified figure comparison base heuristic rollout algorithm heuristic rollout algorithm heuristic multiple scenarios objective ficing numerous candidate locations case limited epn tree search minimal impact performance rollout algorithm approximating small subset small part entire rollout algorithm explanation behavior suitably explained shows histogram values multiple scenarios result application string actions due heuristic clearly rollout algorithms show significant improvement problem shows simulation result multiple scenarios objective optimized smart base heuristic considered instead simulation study highlights extremely interesting behavior exhibited rollout algorithm initial phase algorithm competes rollout algorithm slightly outperforming rollout algorithm many instances however period days rollout algorithm heuristic comprehensively outperforms since rollout lookahead property offers conservative repair decisions initially despite staying competitive anticipation overcoming loss suffered due initial conservative decisions optimizing foresight emergent behavior exhibited optimization methodology offer significant advantages critical scenarios figure cumulative moving average plot objective base heuristic rollout algorithm simulation results shown used base heuristic base heuristic one used simulations shown case particularly well tuned case despite algorithm shows stark improvement repeated application rollout algorithm shows rollout algorithm random selection candidate damaged locations significantly outperforms objective rollout case repair action optimization epn household units retailers difficult come meaningful heuristics reduce size multiple figure comparison base heuristic rollout algorithm figure comparison base heuristic rollout algorithm multiple scenarios objective string actions provided algorithm computational efforts provide brief description computational efforts undertaken optimize problem huge combinatorial nature assignment problem furthermore simulated multiple damage scenarios different simulation result provided work modeling optimization problem solution methodology implemented matlab rollout algorithm gives multiple calls base heuristic function searches string actions sets akin giving millions calls simulator therefore implementation code achieve run time order micro seconds millions calls simulator possible single call simulator defined evaluating repair action completing rollout process damaged components repaired complete rollout despite use best software practices mitigate large action spaces coding fast simulator imperative match good hardware capability simulations humongous size possible parallelize simulation process fully exploit modern day hardware capability ran simulations summit super computer specifically poweredge haswell nodes utilised node cores case never encountered damage scenario whereas case limited damage scenarios system simulator call run time figure comparison base heuristic rollout algorithm multiple scenarios objective gorithm mean number days multiple damage scenarios provide electricity times total number people days whereas base heuristic similarly shows objective benefit epn recovery per unit time rollout algorithm significantly better provide synopsis results objectives smart base heuristic considered number days reach threshold result algorithm better however note number days achieve objective still lesser using rollout algorithm key inference observation rollout might always significantly outperform smart heuristic never perform worse heuristic shows behaviour similar case might outperform rollout shortterm result myopic long run rollout compressively improves upon managed run simulator calls per node different simulation plots computational efforts emphasize massive combinatorial problems possible achieve performance methodology harnessing powerful present day computational systems implementing sound software practices references references nachlas reliability engineering probabilistic models maintenance methods crc press kochenderfer decision making uncertainty theory application mit press bertsekas bertsekas bertsekas bertsekas dynamic programming optimal control vol athena scientific belmont meidani ghanem random markov decision processes sustainable infrastructure systems structure infrastructure engineering frangopol kallen van noortwijk probabilistic models performance deteriorating structures review future directions progress structural engineering materials ellis jiang corotis inspection maintenance repair partial observability journal infrastructure systems corotis hugh ellis jiang modeling riskbased inspection maintenance cost partially observable markov decision processes structure infrastructure engineering fereshtehnejad shafieezadeh randomized value iteration pomdp enhanced counting process technique optimal management systems structural safety bertsekas tsitsiklis programming edition athena scientific kuhn infrastructure management using approximate dynamic programming journal infrastructure systems medury madanat incorporating network considerations pavement management systems case approximate dynamic programming transportation research part emerging technologies busoniu babuska schutter ernst reinforcement learning dynamic programming using function approximators vol crc press directive critical infrastructure security resilience released february bruneau chang eguchi lee orourke reinhorn shinozuka tierney wallace von winterfeldt framework quantitatively assess enhance seismic resilience communities earthquake spectra buldyrev parshani paul stanley havlin catastrophic cascade failures interdependent networks nature barabadi ayele infrastructure recovery prediction recovery rate using historical data reliability engineering system safety watts strogatz collective dynamics nature mcallister research needs developing riskinformed methodology community resilience journal structural engineering concluding remarks study proposed optimization formulation based concept rollout identification community recovery actions following disaster methodology utilizes approximate dynamic programming algorithms along heuristics overcome curse dimensionality analysis management infrastructure systems shown proposed approach implemented efficiently identify recovery decisions following severe earthquake electrical power serving gilroy different base heuristics used recovery defined show applicability approach intended methodology rollout policies significantly enhanced different base heuristics efficient performance rollout formulation optimizing different common objective functions community resilience demonstrated believe methodology adaptable infrastructure systems hazards acknowledgement research herein funded national science foundation grant support gratefully acknowledged opinions findings conclusions recommendations presented material solely authors necessarily reflect views national science foundation work utilized rmacc summit supercomputer supported national science foundation awards university colorado boulder colorado state university rmacc summit supercomputer joint effort university colorado boulder colorado state university cimellaro solari arcidiacono renschler reinhorn bruneau community resilience assessment integrating network interdependencies association bay area governments city gilroy annex url http harnish housing element policy document background report url http adigaa agashea arifuzzamana barretta beckmana bisseta chena chungbaeka eubanka guptaa generating synthetic population united states kahle wickham ggmap spatial visualization journal board institute practical lessons loma prieta earthquake report symposium sponsored geotechnical board board natural disasters national research council symposium held conjunction earthquake engineering research institute national academies press jayaram baker correlation model spatially distributed intensities earthquake engineering structural dynamics abrahamson silva kamai update prediction equations based data set pacific earthquake engineering research center kennedy ravindra seismic fragilities nuclear power plant risk studies nuclear engineering design colombi borzi crowley onida meroni pinho deriving vulnerability curves using italian earthquake damage data bulletin earthquake engineering singhal kiremidjian method probabilistic evaluation seismic structural damage journal structural engineering jaiswal aspinall perkins wald porter use expert judgment elicitation estimate seismic vulnerability selected building types proc world conference earthquake engineering lisbon portugal sep mrl loss estimation methodology earthquake model department homeland security fema washington xie tang tang xie xue seismic fragility assessment transmission towers via analysis proceedings world conference earthquake engineering lisbon portugal google scholar ouyang min resilience analysis framework urban infrastructure systems structural safety nozhati ellingwood mahmoud van lindt identifying analyzing interdependent critical infrastructure urban reconstruction proceedings eleventh national conference earthquake engineering earthquake engineering research institute los angeles june nozhati ellingwood mahmoud sarkale chong rosenheim approximate namic programming approach community recovery management engineering mechanics institute sarkale nozhati chong ellingwood mahmoud solving markov decision processes recovery via simulation optimization rollout submitted publication invited paper bertsekas rollout algorithms discrete optimization survey springer new york new york url https bertsekas tsitsiklis rollout algorithms combinatorial optimization journal heuristics url https limnios fault trees john wiley sons ouyang madanat optimal scheduling rehabilitation activities multiple pavement facilities exact approximate solutions transportation research part policy practice liu chong pezeshki bounding greedy strategy string optimization decision control cdc ieee annual conference ieee masoomi van lindt restoration functionality assessment community subjected tornado hazard structure infrastructure engineering tesauro galperin policy improvement using search mozer jordan petsche eds advances neural information processing systems mit press url http pdf abramson general model static evaluation ieee transactions pattern analysis machine intelligence ragi mittelmann chong directional sensor control heuristic approaches ieee sensors journal bertsekas lids rollout algorithms constrained dynamic programming
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unsupervised learning videos using temporal coherency deep networks carolina roberto jan gram university spain abstract work address challenging problem unsupervised learning videos existing methods utilize continuity contiguous video frames regularization learning process typically temporal coherence close frames used free form annotation encouraging learned representations exhibit small differences frames type approach fails capture dissimilarity videos different content hence learning less discriminative features propose two siamese architectures convolutional neural networks corresponding novel loss functions learn unlabeled videos jointly exploit local temporal coherence contiguous frames global discriminative margin used separate representations different videos extensive experimental evaluation presented validate proposed models various tasks first show learned features used discover actions scenes video collections second show benefits unsupervised learning unlabeled videos directly used prior supervised recognition tasks actions objects images results show features even surpass traditional heavily supervised plus strategy keywords unsupervised learning deep learning object recognition object discovery introduction numbers speak hours video content added youtube every need certainly arises huge source online videos ability automatically learn ideally without requiring annotation form supervision corresponding author email address carolina roberto url http carolina roberto https preprint submitted arxiv january quadruplet network query image siamese network query image neighboring negative neighboring neighboring negative figure siamese deep learning network configurations unsupervised learning videos models simultaneously exploit local slowness present video sequence global discriminative margin two different video sequences state art approaches learning videos fully supervised means rely video datasets heavily annotated aspect problem far every year novel fully annotated video dataset appears thumos activitynet question unsupervised learning first answer obvious strong supervision model implies tremendous effort annotating data timeconsuming task associated costs irremediably prone biases errors reasons justify need truly unsupervised learning approach especially videos fundamentally videos much higher dimensional entities compared images therefore learn long range structures generally implies lot feature engineering computing right kind flow trajectories features using elaborated feature pooling methods furthermore like pointed every minute hundreds videos freely available without annotations none heavily supervised solutions benefit invaluable source information current unsupervised methods use temporal coherence contiguous video frames regularization learning process way spatiotemporal continuity close frames used free form annotation supervision used learn representations exhibit small differences frames however approaches fail capture dissimilarity videos different content hence learning less discriminative features paper also choose work fully unsupervised learning setting models access unlabeled videos key contributions work summarized follows introduce two novel fully unsupervised learning approaches based siamese convolutional neural network cnn architectures exploit local temporal coherence contiguous frames learn discriminative representation separate videos different content margin models associated novel loss functions detailed section figure shows graphical description approach order demonstrate effectiveness features propose address problems unsupervised discovery actions scenes video collections see section words given set unlabeled videos goal separate different classes actions scenes best knowledge propose two novel experiments two rigorous frameworks objective evaluate models discover different actions scenes using clustering validation metrics according set categories experimental validations presented sections actions scenes respectively finally evaluate unsupervised learning approaches using initialization process different deep learning architectures address supervised tasks human action recognition section object recognition section interestingly results show initialization strategy even surpass standard heavily supervised plus approach related work first approaches unsupervised feature learning videos based dimensionality reduction techniques certainly prior work leveraging video unsupervised feature learning relies concept slow feature analysis sfa essentially idea proposes exploit temporal coherence consecutive video frames form free supervision learn visual representations although long memory lstm based approaches proposed recent work combines cnns sfa strategy introduce regularizer encourage coherence using contrastive hinge loss propose shuffle set consecutive frames train set siamese deep networks detect whether sequence frames correct temporal order feature analysis generalized slow steady proposing model requires sequential features change slowly also change similar manner adjacent time intervals video sequence furthermore objective function combines supervised loss unsupervised regularization term simultaneously learning also used problem dimensionality reduction action recognition finally approaches learn specific representation tracked local patches apply features object recognition problem related mentioned methods aim also learn features unlabeled videos however whereas past work focuses preserving temporal coherence among near frames learning objective first move beyond slowness requiring also frames extracted different randomly selected videos lie close feature space learned technically propose two deep learning models siamese quadruplet cnn architecture learn feature space temporal coherence video sequence kept simultaneously enforce representation frames extracted video much closer random pair frames extracted different videos note approach purely unsupervised combination form supervision objective function allowed contrast also additional steadiness enforced guarantee margin visual representations different videos differences respect also notable first propose work tracked image patches model requires extra patch extraction step using tracking process based matching surf features improved trajectories simply propose use video frames learning model completely finally model extends typical triplet siamese network architecture quadruplet siamese network novel architecture allows enrich traditional sfa methodology extending learning based temporal coherence frames video frames different videos technically introduce novel layer regularization previously explored unsupervised learning videos technically problem consists training cnn feature learning given set unlabeled videos parameters network objective function expressed min weight decay constant encodes set training tuples video frames represents unsupervised regularization loss term following introduce two novel siamese deep learning architectures design two loss functions train networks unsupervised way siamese architecture according sfa paradigm introduced possible learn slowly varying features vectorial input signal videos become ideal candidates input signals simply comprehend temporal coherence contiguous video frames work form supervision free structural sfa encourages following property learned feature space temporally nearby video frames lie close learned representation adjacent video frames one would like following abbreviate assuming learned feature representation function learned parameters network key idea behind first siamese model follows learning want encourage query frame adjacent video frame lie close learned feature space distance query frame randomly extracted frame coming different video higher margin first propose siamese architecture shown figure left consists two base networks share parameters input model pair video frames associated label frames belong video otherwise final output base network feature space note initialize base network weights previous training dataset like instance imagenet technically start random initialization model parameters approach fully unsupervised want network learn feature representation query video randomly chosen different video distance feature representations neighboring frames lower distance query frame frame randomly extracted video propose following contrastive loss function based also exploit negative pairs max distance function follow euclidean distance shown equation contrastive loss penalizes distance pair neighbor frames encourages distance belong video use video labels training therefore could happen randomly selected different video might belong category one plans use learned features classifications tasks might problematic assume little likelihood situation actually happens case consider impose margin frames different videos even belong category pushes feature learning towards representations able capture semantic visual concepts quadruplet siamese architecture previous siamese architecture model able see learned feature representation evolves within video note contiguous frames used capture rich structure visual content changes time propose generalize sfa model introducing feature learning model able impose second order temporal derivative learning video representation using triplets temporally close frames neighbor pair pair negative pair figure quadruplet siamese network key idea query frame adjacent frame lie close learned feature space query frame share similar representation frame frame extracted different video feature representation query frame shown green representations neighbor negative frames appear blue note frames belong video blue green representations closer frames belong different videos consequently proposed architecture able learn embedding video frames feature space semantic preserved frames actions scenes mapped points discovered space model guarantees feature changes immediate future remain similar recent past key idea approach different want guarantee learned feature representation temporal coherence contiguous frames kept distance frames wider temporal locality defined temporal window length always lower distance frames coming different videos summary propose model able capture local slowness video keeping global discriminative representation furthermore objective function combines supervised loss unsupervised regularization term model completely unsupervised propose quadruplet siamese architecture illustrated figure right consists four base networks share parameters input model corresponds tuple four video frames output base network feature space four video frames shown figure want query frame adjacent video frame lie close learned feature space query frame shares similar representation frame video frame extracted different video encoded improved loss function designed train proposed quadruplet architecture follows max represents discriminative global margin discovering categories clustering unsupervised feature learning test video figure unsupervised action scene discovery videos siamese architectures learn fully unsupervised way features used discovery task videos different categories discovered running clustering algorithm features implementation details unsupervised learning approaches apply stochastic gradient descent sgd training solutions built using caffe framework input video frames scaled fixed size pixels base network architectures follows alexnet model convolutional layers stack two fully connected layers layer outputs whose neuron numbers quadruplet network trained siamese model trained architectures share parameters therefore perform forward propagation whole batch single base network calculate loss based output features train models set batch size pairs tuples frames siamese quadruplet networks respectively learning rate starts deep architectures siamese network quadruplet network global discriminative margin fixed discovering actions scenes videos figure illustrates designed approach discovering actions scenes videos given set unlabeled videos goal separate different actions scenes represent take inspiration object discovery approach still images introduced essentially generalize evaluation procedure problem description discovery objects image collections unsupervised discovery actions scenes videos consider ideal fully unsupervised scenario adequate experimental validation unsupervised feature learning approaches introduce paper particular design two experiments two video datasets one discovering scenes dataset discovery actions details experimental setups given section shown figure consider video frame characterized feature vector learned one architectures unsupervised way feature vectors frames videos proceed discover different scenes actions state art discovering objects image collections reached clustering based methods propose solve described tasks clustering algorithm learned features note pipeline generalized partition technique built partitions time evaluate performance discovery tasks technically video frames separated set clusters therefore propose use standard metrics scoring discovery quality clusters used particular conditional entropy conditional entropy follow classical formulation proposed information theory consider ground truth category labels obtained cluster labels given variables sampled finite discrete joint space define follows log nutshell experiments consists following steps first set training videos used learn proposed architectures unsupervised way see section second proceed compute test frames test videos corresponding feature representation test features obtained run clustering algorithm evaluate discovery performance note point must fix number clusters build clustering algorithms ultimately machine able decide without human intervention want define clear experimental evaluation different models compared given known number classes note perform parameter tuning experiment simply repeated times average performance standard deviation reported according metric experiments designed experiments accomplish following targets get quantitative qualitative understanding unsupervised models discover scenes actions videos evaluate whether learned representations able generalize well problem changes experimental setup discover actions videos propose use dataset action recognition dataset realistic action videos collected youtube consists videos categorized human action classes dataset divided three splits experiments use consists videos training testing table comparison different unsupervised approaches using dataset lower better metric reported alexnet alexnet alexnet alexnet methods discovering scenes videos chosen publicly available dataset released consists multiple egocentric videos acquired single user different personal contexts car coffee vending machine office home office four different wearable cameras used recording scenes assuming user turn head move body interacting environment follow splits provided dataset training set consists videos seconds test set dataset provides medium length videos minutes normal activity user corresponding personal context although datasets come annotations videos note use video annotations provided learning features instead train unsupervised models subset random video frames extracted videos technically models learned collecting pairs tuple frames randomly sampled training videos discovering actions let use challenging dataset offer detailed analysis proposed models start proposing pipeline perform discovery action categories performance siamese quadruplet architectures evaluated using two different clustering algorithms learned features spectral clustering follow normalized cuts spectral clustering using gaussian kernel similarity matrix also investigate impact learned features obtained different layers cnns layers results summarized table table comparison siamese architecture quadruplet model reveals latter always better also interesting observe features best performance offering come layer instead clustering methods architectures finally results show able discover slightly better categories offer table comparison models competing methods first one consists random assignment video frames clusters baseline random table first row baseline random provides sanity check methods always perform better case note randomly assigning video frames clusters results value close maximum giving idea difficulty action discovery table unsupervised discovery actions dataset comparison baselines lower better metric reported baseline random alexnet lsf alexnet alexnet methods task proposed dataset competing method propose follows standard sfa implementation based contrastive loss technically baseline implemented identical siamese architecture alexnet instead using loss contrastive loss used contiguous frames coming videos following consider comparison important form ablation study empirically validate benefits using novel loss functions incorporate concept global margin different videos siamese model features able outperform dataset performance sfa approach reporting superior respect sfa however best performing method quadruplet network clearly outperforms sfa baseline justifies necessity incorporating learning model global margin features different videos also capability model measure content changes time within video global margin guaranteed letting temporal locality features video play role overall applying unsupervised procedure remaining uncertainty true action category reduced random assignment quadruplet network unsupervised models able compete standard fully supervised approaches used learning features compare performance fully unsupervised solutions performance reported features extracted using fully supervised alexnet model trained imagenet reported layer features obtained fully supervised model respectively best quadruplet network trained roughly videos fully unsupervised way reports means unsupervised solution need need annotation able recover heavily supervised performance defines new state art fully unsupervised training action discovery finally proceed perform qualitative analysis results figure shows random images top three clusters discovered models including fully supervised solution first one observe quadruplet network able obtain distribution similar obtained alexnet trained imagenet moreover figure also shows solutions able build clusters alexnet trained alexnet trained fully supervised alexnet trained imagenet figure qualitative results dataset random images top three clusters discovered shown almost without form supervision feature learning stage figure show embedding using features learned quadruplet architecture words arranges images similar cnn code nearby embedding one observe fully unsupervised solution able learn features actually discover groups semantically similar images discovering scenes experimentally validate detailed models different discovery task propose use dataset discovering scenes personal context videos proceed train feature learning solutions fully unsupervised way based solutions evaluated using features layer according previous analysis dataset features perform better case dataset well table shows detailed analysis performance approaches well comparison random assignment labels standard sfa based contrastive loss note random baseline achieves dataset value close maximum number classes first architectures significantly outperform random baseline second performance figure embedding features learned quadruplet model best viewed zoom unsupervised models better achieved sfa approach detailed comparison observe also performance siamese architecture slightly better one reported sfa model remember case experiments dataset unsupervised models learn egocentric videos scenes activities largely defined objects camera wearer interacts therefore appearance variability personal videos high different users interact objects coffee machine aspects clearly benefit temporal coherency networks analyze results two architectures observe quadruplet model clear winner competition reporting lowest clustering algorithms applying quadruplet architecture remaining uncertainty true context category reduced random assignment one time results reveal benefits using loss takes advantage jointly guaranteeing slowness video table comparison different unsupervised approaches using dataset lower better metric reported methods baseline random alexnet lsf alexnet alexnet frames features discriminative representation worth compare performance unsupervised solutions heavily supervised feature learning model alexnet using imagenet dataset using clustering supervised features reported excellent result explained fact already pointed scenes egocentric videos defined objects user interact therefore alexnet model trained object classification report good performance end experiment showing qualitative results figure figures show main confusion models scenes home office office overall approaches seem able discover finer granularity expected joining video frames containing type screen object category shared across three different scenes office home office even including cluster images windshield confused monitor unsupervised features generalize well present additional experiment aim evaluate whether learned visual representations able generalize well idea simple propose specifically measure performance deep architecture trained video frames discovering scenes test videos dataset evaluate quadruplet siamese alexnet network use clustering discovery task surprisingly able report number clearly outperforms previously best fact reveals models clearly benefit using videos learning feature encoding actually heterogeneity videos training better capability features models discover categories unlabeled video prior supervised recognition section explore benefits initializing deep networks supervised learning task weights provided fully unsupervised strategies typically standard recipe obtaining good performance novel recognition task using deep learning models suggests start initializing weights networks pretrained model learned heavily supervised way table results pascal voc action dataset comparison using cnn architectures models accuracy baseline random assignment alexnet init random alexnet init pascal voc alexnet init pascal voc alexnet init lsf pascal voc alexnet init pascal voc alexnet init trained data pascal voc fully supervised alexnet pretrained imagenet using large scale dataset imagenet second step consists process adapt network new problem task contrary suggest change initialization strategy instead using weights fully supervised model simply propose initialize models following unsupervised learning methodology section present detailed experimental evaluation compare two initialization strategies two different recognition tasks action recognition object classification experiments experimental setup two datasets used proposed tasks action recognition evaluated pascal voc dataset strictly follow procedure detailed pascal voc dataset offers action categories images cropped using action annotations provided images classes images per class used stage test images randomly extracted validation set object recognition experiment use dataset consists color images classes images per category follow experimental setup proposed use training images stage use testing images test models video datasets use perform fully unsupervised learning former random pairs tuples frames extracted videos provided split dataset videos clips randomly selected used extract frames unsupervised feature learning action recognition start experimental evaluation training unsupervised models videos dataset use learned model weights initialize alexnet network action recognition performed table summarizes main results first report performance two baselines random assignment action categories sanity check random initialization weights network interestingly second baseline gets action classification accuracy report performance two unsupervised feature learning models siamese loss quadruplet loss additionally report performance siamese model implement sfa model contrastive loss function clearly quadruplet solution outperforms baselines large margin also siamese models table also shows comparison models able learn deep models unlabeled videos experiments used available models model trained using also dataset exactly videos use models however model uses million triplets frames extracted youtube videos using urls provided use videos overall quadruplet network able improve achieve best accuracy recall model needs work tracked patches requires extra tracking process needs computation local features surf costly improved trajectories models simply learn video frames completely fashion finally table shows accuracy obtained fully supervised alexnet model pretrained using whole imagenet used best unsupervised solution able recover accuracy fully supervised solution consider results promising indicating models able learn effective visual representations action recognition task moreover videos used learning improve defined also want evaluate performance models experiment described experimental setup proposed consists initializing cifar cnn architecture using unsupervised models feature learning videos clips randomly extracted dataset task action recognition pascal voc performed compare following model parameters initialization strategies random assignment using weights obtained unsupervised feature learning models siamese architecture trained sfa contrastive loss approach implementing heavily supervised pipeline initially training model recognize classes dataset table summarizes main results first siamese quadruplet solutions outperform standard random initialization baseline also initialization using sfa model like previous experiments quadruplet model improves siamese architectures excellent work reports best performance slightly improves accuracy heavily supervised pipeline accuracy quadruplet model means able recover accuracy fully supervised model fact heavily supervised model needs table results pascal voc action dataset comparison using cnn architectures models accuracy cifar init random cifar init pascal voc cifar init pascal voc cifar init lsf pascal voc cifar init pascal voc cifar pretrained images cifar pretrained full table object recognition experiment accuracy comparison dataset models accuracy cifar init random cifar init cifar init least training images report accuracy closer consider numbers promising results technically experiments show features learned unlabeled video even surpass standard heavily supervised approach recognizing object categories want conclude experimental validation presenting type experiment problem object recognition use dataset image classification done training images dataset table compare classification accuracy three different architecture initialization strategies random initialization weights approach best unsupervised model quadruplet architecture implemented ranking euclidean loss described using architecture adapted cifar network provided caffe name approach cifar init table model outperforms random initialization showing also feasible learn effective visual representations problem object recognition unlabeled videos remarkably training conditions quadruplet siamese model slightly outperforms learning procedure number provided authors conclusions paper introduce two novel cnn architectures unsupervised feature learning unlabeled videos form jointly exploit local temporal coherence contiguous frames global discriminative margin used separate representations different videos using four diverse datasets show impact methods activity scene discovery tasks activity object recognition problems action classification problem experiments show unsupervised models even surpass heavily fully supervised approach overall consider promising results reported confirming learning unlabeled video augment visual learning plan publicly release code needed reproduce results paper acknowledgments work supported project prepeate reference spanish ministry economy industry competitiveness gratefully acknowledge support nvidia corporation donation titan pascal gpu used research cloud computing resources kindly provided microsoft azure research award references references yang convolutional neural networks human action recognition pami karpathy toderici shetty leung sukthankar largescale video classification convolutional neural networks cvpr simonyan zisserman convolutional networks action recognition videos nips herath harandi porikli going deeper action recognition survey image vision computing kuehne jhuang garrote poggio serre hmdb large video database human motion recognition iccv soomro zamir shah dataset human actions classes videos wild idrees zamir jiang gorban laptev sukthankar shah thumos challenge action recognition videos wild cviu escorcia ghanem niebles activitynet largescale video benchmark human activity understanding cvpr wang oneata verbeek schmid robust efficient video representation action recognition ijcv turchini seidenari del bimbo understanding localizing activities correspondences clustered trajectories cviu fernando gavves oramas ghodrati tuytelaars rank pooling action recognition pami jayaraman grauman slow steady feature analysis higher order temporal coherence video cvpr misra zitnick hebert shuffle learn unsupervised learning using temporal order verification eccv mobahi collobert weston deep learning temporal coherence icml srivastava mansimov salakhutdinov unsupervised learning video rrepresentation using lstms icml van hateren ruderman independent component analysis natural image sequences yields filters similar simple cells primary visual biological sciences royal society hurri hyvarinen receptive fields maximize temporal coherence natural neural computation wiskott sejnowski slow feature analysis unsupervised learning neural computation hadsell chopra lecun dimensionality reduction learning invariant mapping cvpr sun jia chan fang wang yan slow feature analysis action recognition cvpr wang gupta unsupervised learning visual representations using videos iccv zou unsupervised learning visual invariance temporal coherence nips workshop deep learning unsupervised feature learning bay ess tuytelaars van gool surf speeded robust features cviu wang schmid action recognition improved trajectories iccv jia shelhamer donahue karayev long girshick guadarrama darrell caffe convolutional architecture fast feature embedding arxiv preprint krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips tuytelaars lampert blaschko buntine unsupervised object discovery comparison ijcv furnari farinella battiato recognizing personal contexts egocentric images iccv workshop third international workshop assistive computer vision robotics jordan weiss spectral clustering analysis algorithm nips van der maaten accelerating using algorithms journal machine learning research everingham van gool williams winn zisserman pascal visual object classes challenge results krizhevsky learning multiple layers features tiny images tech university toronto liang liu wei liu lin yan towards computational baby learning approach object detection iccv car homeoffice homeoffice office office office car car car car car car homeoffice homeoffice homeoffice homeoffice homeoffice office office office office homeoffice homeoffice homeoffice office office office office office office office office office office coffeemachine coffeemachine coffeemachine car car coffeemachine coffeemachine coffeemachine homeoffice alexnet trained car homeoffice office car car car car car homeoffice coffeemachine coffeemachine coffeemachine coffeemachine homeoffice homeoffice homeoffice office office office car car coffeemachine coffeemachine coffeemachine coffeemachine office office car car coffeemachine homeoffice office office office homeoffice office alexnet trained coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine coffeemachine car car car car car car car car car car homeoffice homeoffice homeoffice homeoffice homeoffice homeoffice office office office office homeoffice homeoffice homeoffice homeoffice office office office office office office fully supervised alexnet trained imagenet figure qualitative results dataset random images categories discovered
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journal communication information systems vol methods block fading mimo channels nov ricardo bohaczuk venturelli danilo silva forcing alternative approach conventional linear receivers systems receiver integer linear combinations messages extracted received matrix individual message recovered recently approach generalized block fading scenario among existing variations scheme ones highest achievable rates drawback efficient algorithm known find best choice integer linear combination coefficients paper propose several methods find coefficients low complexity covering parallel successive interference cancellation versions receiver simulation results show proposed methods attain performance close optimal terms achievable rates given outage probability moreover implementation using root ldpc codes developed showing benefits proposed methods also carry practice index fading linear receivers lattices root ldpc codes successive interference cancellation ntroduction receivers alternative conventional methods equalization mmse equalization mimo channels approach follows framework relay networks receivers attempt extract integer linear combinations transmitted messages received signals recovering messages although joint maximum likelihood receivers achieve best performance among methods searching possible transmitted codewords complexity prohibitively high increasing exponentially number users contrast receivers much lower complexity approach performance many situations moreover performance receiver improved situations successive computation leading successive sif receiver analogously successive interference cancellation sic technique conventional linear receivers hoc associate editor coordinating review manuscript approving publication cristiano panazio ricardo bohaczuk venturelli danilo silva department electrical eletronic engineering federal university santa catarina brazil work partially supported capes grant cnpq grants preliminary version paper presented xxxiv brasileiro processamento sinais sbrt brazil august digital object identifier recent works include design practical channel codes compatible efficient methods select coefficients integer linear combinations well application relay networks moreover principle used receive method mimo uplink scenario also precoding mimo downlink scenario even source coding main results receivers consider static fading symbols codeword subject channel fading however practical situation powerful code large blocklength used may realistic assume symbols codeword subject channel fading therefore channels allow block fading channel fading vary transmission codeword seem realistic model recent work bakoury nazer generalize approach block fading scenario described two decoding methods block fading called arithmetic mean geometric mean decoding decoding method approximates effective noise seen blocks equalization variance case static fading hand decoding method optimally exploits diversity inherent channel variation allowing achieve higher rates decoding decoding methods applicable sif receivers rates achievable receivers depend choice integer matrix specifying coefficients linear combinations decoded however finding optimal choice appears hard problem efficient approximation algorithm known making receivers currently infeasible implement practice nevertheless results obtained exhaustive search demonstrate optimal significantly outperform terms achievable rates outperforms main contribution paper develop lowcomplexity optimization methods receivers gmif closely approach performance obtained optimal search proposed method based optimizing tractable lower bound achievable rate performed efficiently simulation results show proposed method small gap compared optimal performance remaining two methods least scenarios tested performance almost indistinguishable optimal one another contribution paper develop journal communication information systems vol complexity implementation receivers using finitelength codes constellation order validate results practical constraints attaining performance requires use fulldiversity codes adopted root ldpc codes simulation results shown consistent theoretical ones expected gap due finite codeword length remainder paper organized follows system model described section section iii reviews integer forcing static fading well block fading section reviews successive integer forcing fading types section present proposed methods selecting section describes implementation practical constraints simulation results shown section vii covering performance well performance practical codes lastly conclusions presented section viii notation define max log denote row vectors lowercase bold letters matrices uppercase bold letters vector denoted kxk matrix denotes transpose use respectively denote identity matrix appropriate size always clear context set matrices entries set denoted ystem odel consider real gaussian mimo multipleaccess channel mac transmitters one receiver subject block fading independent fading realizations per codeword specifically let codeword length assume simplicity divides let denote vector transmitted transmitter represented row matrix rnt similarly let rnr matrix whose jth row represents vector received jth receive antenna assuming fading realizations equal length received matrix expressed assume receiver perfect knowledge channel realization transmitters knowledge aware channel statistics transmitted vector assumed encoding message produced transmitter message space encoder rate transmitters defined moreover transmitted vectors must satisfy symmetric power constraint assumptions equal power equal rate reasonable since transmitters knowledge channel matrix convenience denote snr receiver decoder attempts recover messages producing estimates error said occur error probability scheme pair denotes error probability fixed channel realization fixed rate said achievable sufficiently large exists scheme rate least given family schemes indexed let rscheme denote maximum achievable rate fixed channel realization target rate outage probability defined pout rscheme fixed probability outage rate defined rout sup pout remark adopted channel order facilitate comparison optimal methods require exhaustive complexity becomes prohibitively large channel even low well existing literature mostly considers channels loss generality since always possible express channel channel alternatively expressions presented straightforwardly generalized complex case replacing transpose conjugate transpose rnr matrix channel fading coefficients ith block rnt rnr gaussian noise matrix entries zero mean variance note remains constant throughout transmission block symbols vary independently realizations convenience let iii nteger orcing aid understanding first review integer forcing static fading describing extension block fading simplicity superscript indicating block index omitted static fading receiver shares basic structure conventional linear receiver illustrated fig first received matrix linearly transformed journal communication information systems vol ynr siso decoder siso decoder siso decoder conventional linear receiver variance vector zeff zeff mth row zeff respectively optimal equalization matrix given integer matrix found using mmse estimation aht hht fig receiver ization matrix resulting stream individually processed channel decoder difference case receiver equalizer attempts estimate transmitted signals directly rather integer linear transformation transmitted signals inverted noise removed crucial noise removal step channel decoders use lattice code common transmitters lattice discrete subgroup closed integer linear combinations particular lattice expressed generator matrix follows every words since also codeword code decoder able decode denoise regardless value chosen detail consider received matrix integer matrix receiver let applies equalization matrix rnt create effective channel output yeff bhx zeff zeff zeff effective noise integer linear transformation assuming chosen lattice row also lattice point thus decoding successful recovered also recovered provided consequence shown following rate achievable rif min log one might think stricter condition required namely invertible would true element could chosen transmission without power constraint however nested lattice shaping used satisfy power constraint possible show requiring rank enough see details case mat achievable rate becomes rif min snr mat optimizing choice achievable rate rif max rank rif seen optimal matrix solving consists set linearly independent integer vectors minimizing since symmetric positive definite admits cholesky decomposition ggt rnt leading kam thus optimal solution described integer coefficients basis set shortest linearly independent vectors lattice generated known shortest independent vector problem sivp believed however suboptimal algorithms basis exist find approximation polynomial time lll algorithm note chosen scheme reduces mmse equalization thus integer forcing generalizes linear equalization providing potentially higher achievable rates block fading case block fading complication arises since block transmitted matrix experiences different channel realization possible independently equalize block received matrix resulting integer linear transformation must blocks order rows remain lattice codewords special case optimal basis found efficiently using lagrange algorithm journal communication information systems vol precisely ith block let znt rnt corresponding integer equalization matrices respectively compute effective channel output block yeff zeff zeff zeff follows yeff zeff yeff zeff denote horizontal concatenation yeff zeff respectively however see general rows guaranteed lattice points since generally express integer linear transformation instance first half codeword concatenated second half another codeword guaranteed form codeword general thus integer forcing work block fading channel require equal given optimal equalization matrix ith block obtained aht snr variance vector zeff row zeff given mth row let zeff zeff zeff denote mth row zeff since noise variance may different block achievable rate expression immediate obtain may fact depend type decoder used specifically depends whether decoder properly exploits diversity inherent block fading channel two decoding methods proposed analyzed arithmetic mean geometric mean decoders discussed decoder decoder attempt exploit channel variation instead treats component effective noise vector zeff variance denoted variance given arithmetic mean hence decoder name variance effective noise block eff shown receiver achieves following rate min log snr min log mam mam thus maximum rate achievable method max rank note identical replaced mam thus solved way case static fading particular lll algorithm similar may used find approximately optimal solution polynomial time special case consists choosing corresponds conventional mmse equalization followed decoder scheme referred achievable rate denoted decoder decoder optimally exploits fact effective noise variance constant across blocks rate achievable method understood average achievable rate among individual blocks could treated parallel channels clearly upper bound precisely rate achievable proven log min min log unless code expressed cartesian product shorterlength codes effectively violates assumption block fading corresponding static fading shorter codeword length note geometric mean hence decoder name variance effective noise block journal communication information systems vol therefore maximum achievable rate max rank useful pointing matrix decoder always achieves rate least high decoder since due inequality however currently known efficient method find optimal approximately optimal solution making optimal currently infeasible implement practice especially grows special case consists choosing corresponds conventional mmse equalization followed decoder scheme referred achievable rate denoted note generalized covariance matrix identity matrix effective channel output rewritten yeff since lower triangular matrix yeff denotes entry note yeff affected decoder acts row yeff successive way suppose successfully recovered compute remove influence second row yeff uccessive nteger orcing one way improve performance conventional linear receiver apply successive interference cancellation sic codeword successfully decoded receiver use side information order cancel part interference reducing variance effective noise principle applied well specifically successive sif receiver integer linear combination codewords successfully decoded used cancel contribution effective noise affecting remaining linear combinations potentially enabling higher achievable rate note sif decoding must done sequentially contrast conventional may done parallel thus decoding order relevant assume decoding follows index thus contrast conventional ordering rows may impact achievable rates sif start reviewing sif static fading followed extension block fading decoded presence less noise generalizing suppose successfully recovered providing estimates remove influence noise obtain yeff decoded noise variance corresponding noise vector estimated proceeding way shown following rate achievable rsif min log choosing optimal maximum achievable rate static fading rsif recall effective channel described section iii sif receiver exploits fact rows zeff correlated general starts performing whitening transformation consider generalized covariance matrix zeff defined kzeff zeff eff assuming optimal equalization matrix used possible show kzeff amat defined since kzeff symmetric positive definite matrix admits cholesky decomposition kzeff llt lower triangular matrix strictly positive diagonal entries define zeff max rank rsif however finding optimal matrix sif decoder different harder problem decoder particular row permutation may give different achievable rate shown possible restrict choice class unimodular matrices optimal solution obtained finding basis lattice generated rnt obtained cholesky decomposition ggt finding basis lattice involves finding shortest lattice vector therefore problem suboptimal algorithms used example applying lll algorithm successive times iteration dimension underlying lattice decreases note possible choose denotes row permutation identity matrix case method reduces conventional sic decoding referred principle possible permutations journal communication information systems vol could tested however quickly becomes unattractive number users increases heuristic methods applied find good decoding order instance decoding first user highest snr lowest expense performance degradation block fading case parallel successive block fading scenario required remain blocks rows still lattice points lattice decoding subsequent inversion possible steps static fading applied separately ith block namely equalization given cholesky decomposition kzeff zeff eff defined successive noise cancellation estimation yeff note effective channel cancellation expressed simply horizontal concatenation respectively particular ith block reduced effective noise possibly different variance lattice decoding step either two decoding methods discussed namely decoding may used generalization case block fading straightforward decoder decoder treats white noise variance follows achievable rate given min log max rank note due inequality matrix decoder achieves rate least high decoder maximum achievable rate given max rank however currently known efficient method find optimal approximately optimal solution making optimal infeasible implement practice even exhaustive search costly gmif since row permutations must considered special case always possible choose corresponds conventional sic decoder method referred achievable rate given max roposed ethods although higher achievable rates fact efficient algorithm known find even approximately optimal undermine practical applicability methods section propose four suboptimal methods choosing integer matrix first two applicable third applicable fourth applies proposed method let rank however currently known efficient method find optimal approximately optimal choice chosen minimize note choose method reduces conventional sic decoder method referred achievable rate given max decoder decoder attempts optimally exploit variation noise statistics across blocks achievable rate computed average achievable rate blocks given snr log min snr min log arg max optimizing becomes optimal matrix propose use either matrix identity matrix depending one gives highest rate decoding let rate achievable method given max note matrix aam chosen rate least high achieved since journal communication information systems vol choice decoder always outperforms decoder hand identity matrix chosen proposed method becomes therefore achieves rate thus proposed method achieves rates high complexity method dominated finding optimal matrix approximated polynomial time lll algorithm therefore complexity proposed method addition choices discussed propose test also optimal matrix ith block would obtained static fading namely arg max rif rank let rate achievable method given max note proposed method chooses matrix may reasonably good blocks simultaneously necessarily optimal block proposed method expands choice including matrices optimal least one block even worse others blocks reasoning behind approach contrast decoder decoder limited performance worst block since individual rates added complexity proposed method higher proposed method since necessary run lll algorithm times since lll algorithm done polynomial time proposed method still viable practice especially small proposed method recall receiver correctly takes account fact effective noise matrix different generalized covariance matrix kzeff block using corresponding noise cancellation lattice decoding step treats reduced effective noise equal variance across blocks upper bound variance obtained treating block effective noise matrix zeff generalized covariance matrix given kzeff zeff eff zeff eff kzeff amat point proceed similarly case static fading section obtaining reduced effective noise variance lower triangular matrix positive diagonal entries given cholesky decomposition kzeff llt follows since suboptimal noise cancellation scheme clear scheme described uses block equalization followed static noise cancellation snc decoding contrast uses block noise cancellation distinguish refer scheme achievable rate given min lower bound let arg max rank optimal matrix propose use matrix rate achievable method given note optimizing exactly problem optimizing thus discussed section computed reduction lattice generated rnt obtained cholesky decomposition ggt procedure approximated applying lll algorithm times proposed method extended ideas proposed method case successive decoding simply redefining sif counterparts using instead proposed method however since decoding order important principle would test permutations identity matrix number permutations grows exponentially number users avoid complexity simply exclude identity matrix permutations set possible choices let asif asif asif arg max rsif rank optimal matrix ith block would obtained sif static fading rate achievable method given max similarly proposed method complexity method grows linearly number blocks however finding well asif requires running lll algorithm times total runs upper bound also derived directly diagonalizing positive definite matrix llt journal communication information systems vol table omplexity nteger orcing ethods method complexity proposed method proposed method proposed method proposed method optimal log log log log log snr summary complexity table shows complexity finding method discussed methods since known priori complexity negligible proposed method complexity corresponding single run lll algorithm note fixed proposed method approaches complexity proposed method successive scenario optimal choice found testing permutations identity matrix proposed method proposed method similar complexity proposed method respectively finally complexity optimal exhaustive search follows bound see also note besides finding receiver operation includes also tasks equalization channel decoding whose complexity scales blocklength identical method category parallel successive thus task finding tends take smaller fraction overall decoding time grows mplementation ractical odes achievable rate results discussed assume use lattice codes asymptotically high dimension asymptotically large constellation however practice code modulation must used section discuss practical lattice encoders decoders receiver may implemented low complexity focus use binary ldpc codes modulation encoding decoding start conventional extension successive straightforward simplify description slight abuse notation consider finite field size denoted subset integers suppose user encodes message codeword linear block code codeword modulated vector constellation computed mod discrete dither vector independent known receiver mod operation applied assumed return number interval note dither vector must used order reduce transmit power constellation constellation could interpreted simply modulation map case simple choice would however using random dither uniformly distributed convenient purposes since shall see makes error probability independent transmitted codeword consequence use dithers transmitted vector lattice point anymore contrast description sections iii however dithers easily removed receiver simple modification results sections precisely let matrices whose row respectively receiver computes effective channel output yeff mod zeff mod zeff mod zeff defined section iii mod note row codeword thus recover effective channel except let yeff zeff denote jth component yeff zeff respectively order use belief propagation decoder necessary compute ratio llr channel output defined yeff llr log yeff denotes conditional probability simplify decoder effective noise component zeff treated gaussian random variable zero mean variance decoding decoding denotes index block jth component belongs however since channel infinite number noise realizations lead given channel output resulting expression yeff yeff exp order simplify llr computation approximate expression keeping largest actually channel required results omitted overview given thus modulo operation must used receiver transmitted vectors satisfy power constraint journal communication information systems vol important parameter characterizing performance code fading channel diversity order defined real llr approximated llr llr error probability decoder block rayleigh fading channel diversity order code known satisfy bound log log snr lim yeff fig llr channel snr term corresponding element nearest yeff euclidean distance follows yeff llr eff eff floor function code rate bits per channel use thus codes achieve full diversity limited binary codes family binary ldpc codes achieve full diversity belief propagation performance close theoretical limits root ldpc codes codes systematic information bits corresponding first positions block matrix satisfying following structure hijk since yeff fig shows exact llr comparison approximation channel snr see input range approximation indistinguishable exact value approximation slightly less accurate input close integer since correspond peak llr values high degree certainty value loss accuracy degrade performance belief propagation decoding procedure easily extended successive replacing yeff zeff decoding decoding code construction practice approaching performance mmse sic sif reception requires suitable decoder also codes allow decoder exploit diversity issue apparent since achievable rate results even based asymptotically good lattices already optimal exploiting diversity designing codes lowcomplexity constraints however far trivial simply llr structure implies information bit ith block one equation relating bits jth block blocks worth mentioning root ldpc codes belief propagation guarantee full diversity information bits thus one interested recovering entire codeword must regenerated information bits decoding vii imulation esults section present simulation results comparing outage rate performance proposed methods optimal performance obtained exhaustive search comparison include performance previously receiver well conventional linear receivers simulations specify outage probability estimated channel realizations realization channel fading coefficients drawn independently gaussian distribution zero mean unit variance optimal matrix obtained exhaustive search matrices vectors whose row exceed fig shows outage rate receivers channel blocks seen proposed method journal communication information systems vol optimal prop prop optimal prop optimal prop rate rate snr snr fig outage rate channel blocks parallel decoding methods successive decoding methods achieve performance close optimal strictly higher maximum particular outage rate performance proposed method within optimal outperforms proposed method hand proposed method appear performance indistinguishable optimal optimal respectively note proposed method outperforms gmsic approximately outage rate much lower performance case fig shows outage rate scenario blocks snr varying number users assuming number receive antennas due complexity exhaustive search grows exponentially performance optimal shown optimal shown seen increases performance proposed methods appears converge approached similarly performance proposed method significantly improves increases outperforming approaching proposed method nevertheless proposed method still outperforms methods visible margin fig considers scenario fig blocks similar observations made except comparatively proposed methods performance worsened improved still significantly outperformed proposed methods successive methods behavior similar case observed except outperforms proposed method achieves smaller gap proposed method one might expect gap vanish large noted due rate limitations constructions codes typically restricted small case lastly figs show rate fer channel respectively using modulation regular code length constructed using technique used simulation seen simulations consistent theoretical results performance gap due small constellation size channel code particular fer proposed methods within theoretical fer within fer successive decoding shown since methods performance similar counterpart benefits sif become salient lattice codes higher spectral efficiency needed design codes however outside scope paper viii onclusions paper propose four suboptimal methods selecting integer matrix reception block fading scenario two applicable two applicable respectively main idea behind methods use matrix optimized scheme simpler objective function approximately optimal solution found polynomial time corresponding simpler scheme proposed scheme gmif best choice among respective counterpart certain static fading solutions found efficiently shown simulations proposed methods gmif achieve outage rates strictly higher best methods regardless number blocks users slightly complex exactly journal communication information systems vol optimal prop prop rate rate optimal prop prop optimal fig outage rate channel blocks snr parallel decoding methods successive decoding methods optimal prop prop rate rate optimal prop prop optimal fig outage rate channel blocks snr parallel decoding methods successive decoding methods observations hold comparison particular outage rates schemes much superior counterparts also show schemes realized practice low complexity finite codeword length constellation constraints simulation results using root ldpc codes found agree theoretical ones confirming superiority comparison authors would like thank islam bakoury bobak nazer valuable discussions interesting avenue future work development lattice codes higher spectral efficiency instance linear codes use construction codes would directly applicable schemes allowing wider interesting operating range venturelli silva integer forcing block fading mimo channels xxxiv brasileiro processamento sinais sbrt aug zhan nazer erez gastpar linear receivers ieee trans inf theory vol doi acknowledgments eferences journal communication information systems vol liu ling efficient integer coefficient search ieee trans wirel vol doi maham hejazi integer transceiver design mimo multipair relaying ieee trans veh vol doi nazer shamai duality integerforcing ieee international symposium information theory jun doi silva pivaro fraidenraich aazhang integerforcing precoding gaussian mimo broadcast channel ieee trans wirel vol jul doi ordentlich erez source coding ieee trans inf theory vol doi knopp humblet coding block fading channels ieee trans inf theory vol doi bakoury nazer impact channel variation snr receivers ieee international symposium information theory isit jun doi fig fer channel using parallel decoding methods blocks zamir lattice coding signals networks cambridge cambridge university press isbn nazer cadambe ntranos caire expanding framework unequal powers signal levels multiple linear combinations ieee trans inf theory vol doi closest vectors successive minima dual prop prop lattices automata languages programming springer optimal berlin heidelberg jul doi prop theo prop theo regev lattices learning errors random linear codes optimal theo cryptography proceedings annual acm symposium theory computing ser stoc new york usa acm doi isbn agrell eriksson vardy zeger closest point search lattices ieee trans inf theory vol doi lenstra lenstra lovasz factoring polynomials rational coefficients math vol doi steele master class introduction art mathematical inequalities cambridge new york mathematical association america apr isbn snr wolniansky foschini golden valenzuela architecture realizing high data rates fig fer channel using parallel decoding methods wireless channel ursi international symposium blocks signals systems electronics conference proceedings cat doi ordentlich erez nazer successive nazer gastpar harnessing interits optimality annual allerton conference ference structured codes ieee trans inf theory vol communication control computing allerton doi doi feng silva kschischang algebraic approach biglieri coding wireless channels ser information network coding ieee trans inf theory vol transmission processing storage new york springer isbn doi tse viswanath fundamentals wireless communication boutros guillen fabregas biglieri zemor cambridge new york cambridge university press jul density codes nonergodic channels isbn ieee trans inf theory vol doi ordentlich erez achieving gains promised forcing equalization binary codes ieee convention uchoa healy lamare structured rootof electrical electronics engineers israel doi ldpc codes techniques channels wireless com network vol doi chae jang ahn park jeong multilevel coding scheme mimo receivers binary codes ieee trans wirel vol doi ding kansanen wang zhang exact smp algorithms linear mimo receivers ieee trans wirel vol doi fer fer prop prop optimal prop theo prop theo optimal theo journal communication information systems vol ricardo bohaczuk venturelli received degree degree federal university santa catarina ufsc brazil respectively electrical engineering currently student electrical engineering federal university santa catarina research interests include channel coding network coding information theory wireless communications member brazilian telecommunications society sbrt since danilo silva received degree federal university pernambuco ufpe recife brazil degree pontifical catholic university rio janeiro rio janeiro brazil degree university toronto toronto canada electrical engineering postdoctoral fellow university toronto polytechnique lausanne epfl state university campinas unicamp joined department electrical engineering federal university santa catarina ufsc brazil currently assistant professor research interests include wireless communications channel coding information theory machine learning silva member brazilian telecommunications society sbrt recipient capes scholarship shahid qureshi memorial scholarship fapesp postdoctoral scholarship
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oct inferring disease causing genes pathways mathematical perspective jeethu devasia priya chandran department computer science engineering national institute technology calicut india email jeethu department computer science engineering national institute technology calicut india email priya objective system level view cellular processes human several organisms captured analyzing molecular interaction networks molecular interaction network formed differentially expressed genes interactions helps understand key players behind disease development functions genes blocked altering interactions would great impact controlling disease due promising consequence problem inferring disease causing genes pathways attained crucial position computational biology research however considering huge size interaction networks executing computations costly review literatures shows methods proposed finding set disease causing genes could assessed terms accuracy perfect algorithm would find along accuracy time complexity method also important high time complexities would limit number pathways could found within pragmatic time interval methods results problem tackled integrating graph theoretical approaches approximation algorithm problem inferring disease causing genes pathways transformed graph theoretical problem graph pruning techniques applied get results practical time randomized rounding efficient approach design approximation algorithm applied fetch relevant causal genes pathways experimentation multiple benchmark datasets demonstrated accurate computationally time efficient results existing algorithms also biological relevance results analyzed conclusions based computational approaches biological data sets disease causing genes corresponding pathways identified multiple disease cases proposed approach would remarkable contribution areas like drug development gene therapy could recognize results biologically index interaction network causal genes dysregulated pathway graph pruning approximation algorithm randomized rounding ntroduction cellular processes mainly governed biomolecules example particular protein function understood mapping interactions biological communications represented using molecular interaction networks present molecular interaction networks various organisms available exploit diverse aims discovering disease causing genes related pathways computational perspective computing information flows complex model expensive input sizes large analyses typically high time complexities therefore proposing algorithms effective computation networks would remarkable impact knowledge gained networks problem inferring disease causing genes dysregulated pathways prime importance huge academic industrial interest potentially useful comprehending underlying system complex diseases suggesting prospective drug targets algorithm augments graph theoretical approaches approximation inferring causal genes dysregulated pathways addressed paper proposed method incorporates gene expression value potentiality predicting diseases genes correlated genes lower risk vice versa experimental analysis state art related works together proposed method also given paper related works based random walk based approach electric circuit model expression quantitative loci eqtl analysis electric circuit model multiple sources sinks fast iterative matrix inversion approximation algorithm based randomized rounding proposed random walk approach shown significant impact problem identifying causal genes underlying pathways based pearson correlation coefficient gene expression values genes transition probability defined starting source node random walker moves node qualified unvisited highest transition probability bearing node among neighbors current node process repeated visits destination node movement possible candidate causal gene largest number visit times largest value probability causal gene taken causal gene identification pathway done tracing path corresponding source node selecting intermediate nodes visited ones according suthram approach results relatively short walks requirement multiple iterations better results proposed new approach based electric circuit model analogous random walk based approach considering network interactions transcription factor interactions electric circuit conductance edge set based correlation expression values target gene solving electric circuit using kirchhoff ohm laws get current node edge causal gene taken gene highest value current flow pathway shortest route target gene causal gene highest total sum currents across interactions edge corresponds interaction paths together give textitpathways entire network based approaches kim suggested electric circuit based approach multiple sources sinks conductance defined average absolute value pearson correlation coefficient gene expression values target gene genes endpoints link system linear equations based ohm law kirchhoff law solved find voltages links thereby calculate current value causal genes taken genes significant amount current flow pathway shortest paths collection maximum current paths pair source sink focusing faster computation voltages nodes suggested previous approach fourth order iterative method fast iterative matrix inversion proposed following calculation voltages nodes causal genes dysregulated pathways identified citeref yoo new approximation algorithm proposed based electric circuit approach fetch causal genes corresponding pathways collection distinct paths reduced network obtained thresholding computed randomized rounding applied get path maximum current flow taken dysregulated pathway corresponding target causal genes members pathways taken causal genes infer reported works literature finding set disease causing genes could evaluated terms accuracy closeness set found actual set execution time assessment observed cases execution time rises accuracy aterials ethods selection target genes candidate causal genes gene expression data breast cancer lung cancer species homosapiens pancreatic cancer species rattus norvegicus utilized experimentation data collected national center biotechnology information ncbi sponsored gene expression omnibus data repository platform details sample information datasets given table disease geo accession sample count case control breast cancer lung cancer pancreatic cancer platform affymetrix human genome array affymetrix human gene array affymetrix rat genome array table dataset details following steps done dataset separately initial data normalized using robust average method remove noise due factors genes statistical significance selected using equal variance followed calculation significant computed genes qvalue selected set differentially expressed genes taken candidate causal genes breast cancer pancreatic cancer considering data size first genes sorting ascending order following filtering based selected candidate causal genes lung cancer gene interaction network molecular interactions downloaded biogrid database candidate causal gene fetched network data filtered select genes differentially expressed genes linked transcription factors network considered target genes set target genes considered set source nodes set genes apart target genes considered set sink nodes selection benchmark data data ncbi sponsored gene database aceview uniprot database cosmic database curated benchmark data breast cancer lung cancer species homosapiens set associated genes corresponding particular disease case species fetched databases fetched genes removing duplicates taken benchmark data disease similarly data ncbi sponsored gene database aceview uniprot database literatures ncbi nature nucleic acid research compiled benchmark data species rattus norvegicus formulation algorithm biological background gene expressed particular level process producing ultimate products called proteins level measured numerical quantity known expression value also gene interacts several genes resulting certain phenotypes advances area computational biology molecular interactions represented network designating nodes edges associations end nodes associations different aspects like physical interactions membership pathway literature reveals fact two nodes may linked together different cases resulting network multiple edges also molecular interaction network consists certain directional links interactions bidirectional tfdna interactions directional phosphorylation events directional expression levels values genes help make distinction healthy disease cases view fact increase decrease expression values genes many diseases healthy individuals particular gene expression value changes two groups healthy affected individuals gene said differentially expressed differentially expressed genes may lead disease state beneficial state definitions set genes gives rise particular disease state termed causal genes set probable genes certain disease termed candidate causal genes set candidate causal genes bound transcription factors referred target genes described section candidate causal genes connected together network due different aspects disease state may developed interference target gene candidate causal gene normal biological functions cell relationship target gene candidate causal gene known dependence dependence target gene candidate causal gene computed pearson correlation coefficient expression values genes genes low values high values crucial molecular interactions consider absolute value similarly edge weight edge molecular interaction network taken average absolute value dependence genes end nodes edge target gene edge weights used measure inferring causal genes corresponding pathways formal notations definitions let set target genes gcc set candidate causal genes set causal genes basic problem find set causal genes gcc gcc known fundamental decision problem gcc gcc gcc candidate causal gene causal gene let recall gcc role disease would determined interactions target genes candidate genes role disease target gene weighted graph network defined source members gcc nodes molecular interactions edges edge weights reflect role genes represented end nodes causing disease may realized point would graphs containing nodes let network target gene gcc defined molecular interactions described next let gcc gcc nodes would multiple edges section molecular interaction network simplified get ordinary graphs using following method weight function defined candidate causal genes interaction three ways outlined section thus multiple edges defined section get substituted single weighted edge genes endpoints edges higher edge weights said role disease causal genes weight path gcc paths candidate causal genes high weight values called dysregulated pathways may multiple dysregulated pathways may recalled interaction networks typically contains multiple dysregulated pathways genes corresponding nodes belonging dysregulated pathways called causal genes gcc exists path gcc candidate causal gene target gene dysregulated pathway description terms high edge weights high weighted paths used adjectives abstract need quantified towards new algorithm problem defined section explained stated without abstract adjectives protein network represented directed graph source vertex nodes defined previously nodes represent genes proteins edges represent molecular interactions obtained reduced eliminating low weighted edges high low decided based threshold defined follows threshold defined set edges removed computed follows edges removed paths high path weights defined earlier candidate causal genes graphs dysregulated pathways vertices belonging dysregulated pathways causal genes given graphs source vertex gcc find gcc gcc gcc threshold value defined equation find dysregulated pathways paths satisfying condition inferred every research work described section attempts find complete set dysregulated pathways however realistic biological networks size sets number paths large practical computation order get results reasonable computation time following heuristic used work biological network higher degree node relevant degree node number neighboring edges degree node denoted deg hence time path explored vertex gcc deg gcc removed minimum degree let node minimum degree updated next path traversed repeat process till integer process removing made efficient terms execution time mapping gcc along degree node deg gcc data structure underlying data structure priority queues property parent node root node parent parent node priority given terms decreasing order degree highest priority assigned vertex ensure vertex least degree removed considering node degree molecular interaction network elements deleted removing root node data structure reorganized bring highest priority node root process specified earlier continues till number nodes time removed adjacent edges also removed thereby reducing complex nature biological networks turn results identification relevant genes corresponding pathways within practical time interval definitions updated follows gcc gcc gcc deg gcc additional characteristic causal genes incorporated apart attributes described equation causal genes tend higher degree nodes dysregulated pathways paths satisfying mentioned conditions deg deg finally motivated approach randomized rounding efficient approach design approximation algorithm done fetch relevant paths terms weight degree deg gcc algorithm domized rounding approach resulted factor total number paths collection distinct paths reduced network total number distinct edges number occurrences randomly selected edge underlying principle follows pis calculated let max dysregulated pathway taken path value members pathway taken causal genes therefore definitions reformed adding following attribute causal genes thereby dysregulated pathways constituted maximum weighted paths randomized rounding final definitions joining mentioned attributes together given gcc gcc gcc deg gcc dysregulated pathways paths satisfying stated conditions deg deg algorithm infer causal genes dysregulated pathways input graphs source vertex sink vertex gcc gcc output implementation validation results comparative study related works literature proposed method performed based accuracy results execution time methods implemented intel cpu machine memory ram capacity using programming languages also effectiveness algorithm explored using reactome pathway database provide biological significance analysis based threshold defined quite critical removing edges effect identifying causal genes dysregulated pathways simulated apart implementing algorithm algorithm without using steps algorithm implemented also algorithm implemented different values defined section datasets iii iscussion esults approximation algorithm graph reduction pseudo code algorithm identify causal genes dysregulated pathways given algorithm algorithm considers molecular interaction network multiple sources sinks source set set target genes sink set set candidate causal genes target genes proof approximation factor given appendix implementation validation results related works literature proposed algorithm implemented specified section multiple datasets breast cancer dataset consists genes gcc consists genes lung cancer dataset consists genes gcc consists genes pancreatic cancer dataset consists genes gcc consists genes edges nodes total graphs breast cancer dataset edges nodes total graphs lung cancer dataset edges nodes total graphs pancreatic cancer dataset benchmark data specified section obtained cases breast cancer lung cancer pancreatic cancer genes genes genes respectively let denote breast cancer dataset dataset lung cancer dataset dataset pancreatic cancer dataset dataset iii simulation study based effect identifying causal genes dysregulated pathways simulated considering multiple graphs source subset sink nodes dataset source subset sink nodes dataset acaca source gcc gcc gcc calculate find remove repeat indpath goto step else find goto step else find total number distinct edges randomly pick edge path calculate number occurrences randomlyp select path calculate end path value find end end end subset sink nodes dataset iii specified section results summarized tabulated table dataset iii causal genes execution time sec sec sec sec sec sec table effect order without slight increase number causal genes identified method without using compromises execution time considering genes analysis based algorithm implemented different values defined section datasets considering size datasets dataset subgraph considered selecting first genes sorting resulting graphs edges nodes benchmark data consisting procedure indpath calculate deg gcc gcc assign priority based decreasing order degree make min heap find delete min heap update else neighbours adj adj unexplored indpath adj end end end end procedure genes characteristic roc analysis used compare performance algorithm various values values selected roc curve plots true positive rate tpr sensitivity versus false positive rate fpr tpr percentage causal genes correctly identified based benchmark dataset fpr percentage causal genes present benchmark dataset compare different curves obtained roc analysis area curve auc curve calculated given table iii higher value auc better result also execution time case taken summarized table roc curve execution time compared shown better performance selected value algorithm provided overall accuracy cases values dataset dataset dataset iii table iii auc different values values dataset dataset dataset iii table execution time seconds different values fig comparison based accuracy results measuring accuracy results analysis based execution time accuracy defined measurement closeness benchmark data actual results calculated percentage causal genes correctly identified based benchmark dataset comparison based accuracy results given figure execution time given table approaches random walk approach eqed approach current flow approach matrix inversion approach randomized rounding approach rounding dataset min min min min min min dataset sec sec hours hours hours sec dataset iii min min min min min min table comparison based execution time observations made based results given next random walk approach pearson correlation coefficient gene expression levels genes node target gene calculated transition probability obtained using value random walk based transition probabilities done multiple times disease causing genes including pathways obtained result random walk approach takes less amount execution time number genes identified accuracy results less molecular interaction considered electric circuit according approaches literature conductance link calculated based pearson correlation coefficient gene expression levels genes node target gene kirchhoff current law ohm law utilized obtain voltages current calculated using ohm law edge network disease causing genes dysregulated pathways identified using methods eqed approach takes least amount execution time gives lesser number genes low accuracy compared methods random walk approach electric circuit approach multiple sources sinks fast iterative matrix inversion return number disease causing genes accuracy genes receiving current least maximum current among genes considered iteration thereby execution time greater cases since fast iterative matrix inversion uses fourth order iterative method execution time less compared electric circuit approach multiple sources sinks approximation algorithm outperforms methods terms number disease causing genes identified accuracy results takes much time compared random walk approach eqed approach applying randomized rounding possible paths identified turn results increased execution time proposed algorithm approximation algorithm graph reduction implemented multiple datasets sets causal genes associated pathways disease cases identified initially set lessweighted edges removed set vertices removed together exploration set distinct simple paths making sure vertex least degree deleted data structure utilized time efficient vertex removal process repeats times value finally relevant paths identified using randomized rounding newly proposed approximation algorithm graph reduction defeats methods identifying number genes within less time interval sets newly identified disease causing genes apart known genes benchmark datasets well resultant sets identified causal genes dysregulated pathways datasets given supplementary material onclusion uture scope computationally intensive process identifying causal genes related pathways huge molecular interaction networks addressed paper huge size molecular interaction network reduced retaining relevant nodes edges terms connectivity appropriate nodes thereby paths terms parameters selected using concept approximation experimentations proved newly proposed approximation algorithm graph reduction outperforms methods identifying number genes within lesser time eferences kim jozef przytycki przytycka modeling information flow biological networks physical biology iop publishing ltd may winslow leandersson larsson prognostic stromal gene signatures breast cancer breast cancer research vol cho kim chapter network biology approach complex diseases plos computational biology vol wang arbeitman chen sun integrative approach causal gene identification gene regulatory pathway inference bioinformatics vol doyle snell random walks electric networks washington mathematical association america suthram andreas beyer yonina eldar eqed efficient method interpreting eqtl associations using protein networks molecular systems biology vol kim wuchty identifying causal genes dysregulated pathways complex diseases plos computational biology vol jose chandran fast iterative method modeling information flow biological networks master thesis national institute technology calicut india devasia chandran towards improved algorithm modeling information flow biological networks interanational conference advances computing communications information science elsevier irizarry hobbs collin exploration normalization summaries high density oligonucleotide array probe level data biostatistics vol maglott ostell pruitt tatusova entrez gene genecentered information ncbi nucleic acids research vol suppl forbes beare gunasekaran leung bindal boutselakis ding bamford cole ward kok jia teague stratton mcdermott campbell cosmic exploring world knowledge somatic mutations human cancer nucleic acids research vol sandrine dudoit yee hwa yang speed statistical methods identifying differentially expressed genes replicated cdna microarray experiments statistica sinica vol kawasaki end microarray tower babel universal standards lead way journal biomolecular techniques vol millenaar okyere may van zanten voesenek peeters decide different methods 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journal machine learning research submitted published operational calculus programming spaces sajovic corresponding author feb university ljubljana faculty computer information science pot ljubljana slovenia xlab pot brdom ljubljana slovenia martin vuk university ljubljana faculty computer information science pot ljubljana slovenia editor abstract paper develop operational calculus programming spaces generalizes existing approaches automatic differentiation computer programs provides rigorous framework program analysis calculus present abstract computing machine models automatically differentiable computer programs computer programs viewed maps finite dimensional vector space called virtual memory space extend tensor algebra dual accommodate derivatives extended virtual memory algebra programs data structure one calculate elements give expansion original program infinite tensor series program input values define operator differentiation programming spaces implement generalized shift operator terms powers approach offers powerful tool program analysis approximation provides deep learning formal calculus calculus connects general programs deep learning operators map formulations space equivalence enables generalization existing methods neural analysis computer program vice versa several applications presented notably meaningful way neural network initialization leads process program boosting keywords programming spaces operational calculus deep learning neural networks differentiable programs tensor calculus program analysis introduction programming holds power algorithmic control freedom expression whose abstinence severely limits descriptiveness closed form methods pen paper mathematics thus cementing programming language modern problem solving yet vibrant tradition mathematics existed since dawn mind remains exception combinatorics largely untapped computer science discrete nature physical reality studied analytic means nature digital phenomena studying procedures objects undergoing change virtual space partially attempted hamiltonian monte sajovic martin vuk sajovic vuk carlo methods bayesian predictors girolami calderhead studied manifolds great success using unfortunately impractical methods derivatives distributions course freedom algorithmic expression programming embraced way evaluation algorithmic expressions evaluation symbolic expressions standard analysis lies core dissonance disconnect partially mitigated methods automatic utilized today machine learning engineering simulations etc see baydin yet lack proper formalism model collapses see pearlmutter siskind one tries generalize notions program operating derivatives another program coding required within retaining famed expressive freedom models allowing nested pearlmutter siskind still fail providing algorithms algebra enabling study formulation programs analytic equations existing models remain nothing means calculating derivatives void meaningful algebraic insight lacking true power vast analysis endowed aim paper bridging gap programming analysis left behind previous models generalizing shown views great elephant employing tensor algebra virtual memory constructed whose data structures one calculate section exact construction analytic virtual machine presented capable implementing programming spaces operators acting upon supporting actions multiple memory locations time programming spaces shown subalgebra giving rise symbolic manipulations programs attaining construction algorithmic control operators expanding program tensor series derived section operator program composition constructed section generalizing forward khan barton reverse hogan mode automatic arbitrary order single invariant operator problem nested resolved section theory grants ability choose programs complexity approximations virtual space tensor algebra model virtual memory consists maps tailor made implementation gpu parallelism abdelfattah section functional transformations programs arbitrary function basis derived hardware optimized running sets functions proves useful adapting code hardware branching discontinuity discussed section show analytic virtual machines fully integrate control structures retaining algorithmic control theory grants generalization neural networks general tensor networks section revealing connection programs section historic taylor series modernized section deriving neural tensor series establishing equivalence general tensor networks programming spaces direct implications study understanding differentiable neural computers graves operational calculus equivalence transformations arbitrary programs general tensor networks derived transformation neural tensor series serves great initialization point general tensor network trained seen operational calculus programming spaces process boosting freund converting weaker learner strong one proves fruitful currently neural networks give best results many problems constructs trainable training process adhere operational calculus demonstrated section enabling analytic study theory theory new insights programs demonstrate probe inner structure section revealing actions properties domain react section enables alter data imposing property lacks upon state art methods analyzing neural networks mordvintsev generalized use programming spaces section providing framework analysis machine learning architectures virtual constructs actions virtual space computer programs maps vector space model computer programs maps vector space focus real valued variables type float double state virtual memory seen high dimensional set possible states program memory modeled dimensional real vector space call memory space program computer program memory space described map programming space space maps implemented program programming language definition euclidean virtual machine tuple euclidean virtual machine finite dimensional vector space complete field serving subspace space maps called programming space serving actions memory differentiable maps programs programs let recall multivariate calculus definition derivative let banach spaces map differentiable point exists linear bounded operator lim khk map called derivative map point assume variables interest type float simplicity theoretically field used instead applications field sajovic vuk maps derivative expressed multiplication vector jacobi matrix partial derivatives components map assume remainder section map derivative map linear bounded maps assume dimensional space linear maps isomorphic tensor product isomorphism given tensor contraction sending simple tensor linear map derivative map one consider derivative looking map leads higher derivatives definition higher derivatives let map vector space vector space derivative order map map remark sake clarity assumed definition map well derivatives differentiable points case definitions done locally would introduce mostly technical difficulties let basis basis denote component map according basis terms directional partial derivatives formula differentiable programs want able represent derivatives computer program euclidean virtual machine program euclidean virtual machine three subspaces virtual memory space describe parts memory result program denote standard basis memory space dual basis functions coordinate functions correspond individual locations variables program memory operational calculus programming spaces definition program programming space define input parameter space output space minimal vector subspaces spanned standard basis vectors map defined following commutative diagram prop depend choice element space called free space program variables corresponding standard basis vectors spanning parameter output free space called paramters input variables output variables free variables correspondingly free variables left intact program result value output program depends values input variables consists variables changed program input parameters output values might overlap map called effective map program describes actual program memory ignoring free memory derivative map interest speak computer programs definition automatically differentiable programs program automatically exist embedding space free space program effective map map program automatically order exist program effective map map program automatically also map however derivative program map implemented program since memory space limited able program order calculate save derivatives orders less differentiable programming spaces motivated virtual memory programs sequence vector spaces recursive formula sajovic vuk note sum direct since subspaces naturally isomorphic space recursive formula subspace tensor algebra consisting linear combinations tensors rank less equal construction enables derivatives maps domain codomain putting memory considerations aside propose universal model memory programs definition virtual memory let euclidean virtual machine let tensor algebra dual space call virtual memory virtual computing machine term virtual memory used possible embed certain subspaces memory space making similar virtual memory memory management technique extend program map universal memory space setting component direct sum components zero similarly derivatives also seen maps setting component direct sum others zero differentiable programming spaces let following function spaces function spaces seen sub spaces since naturally embedded derivative operator space smooth maps denote operator image map operator derivative higher order derivatives powers operator applied thus mapping function spaces definition differentiable programming space programming space subspace space spanned called differentiable programming space order elements analytic call analytic programming space spaces naturally isomorphic identified sum operator may defined partially maps well handle case later operational calculus programming spaces higher order programming spaces following theorem theorem infinite differentiability differentiable programming space infinitely differentiable programming space meaning proof induction order claim holds assume component derivative multiindex denote denoting component index denoting component thus induction claim holds corollary differentiable programming space order embedded tensor product function space space multitensors order less equal taking limit consider tensor tensor series algebra algebra formal operational calculus programming spaces corollary may represent calculation derivatives map one mapping operator direct sum operators image order direct sum map value derivatives order evaluated point operator recursive relation used recursively construct programming spaces arbitrary order tensor series algebra completion tensor algebra suitable topology sajovic vuk proposition explicit knowledge required construction proof construction achieved following argument proof theorem allowing simple implementation dictated remark maps constructible using tensor algebra operations compositions programs definition algebra product bilinear map define bilinear product following rule simple tensors extending linearly whole space theorem programming algebra bilinear map infinitelydifferentiable programming space function algebra product defined analytic virtual machine propose abstract computational model virtual machine capable constructing programming spaces machine provides framework analytic study algorithmic procedures algebraic means definition analytic virtual machine tuple analytic infinitely differentiable virtual machine finite dimensional vector space virtual memory space analytic programming space differentiable programming space defines infinitely differentiable virtual machine remark tuple structure tensor series algebra sufficient construct programing spaces linear combinations elements illustrative example implementation analytic virtual machine available github sajovic implementation closely follows theorems derivations paper intended educational guide transitioning automatic theory paper sajovic explaining process implementation accompanies operational calculus programming spaces tensor series expansion exists space spanned set thus expression well coordinates operator written series multiindices operator mapping function spaces also map taking image map certain point may construct map space programs space polynomials using note space multivariate polynomials isomorphic symmetric algebra turn quotient tensor algebra element one attach corresponding element namely polynomial map thus similarly consider completion symmetric algebra formal power series turn isomorphic quotient tensor series algebra arriving element expression map mapping program formal power series express correspondence polynomial maps given multiple contractions possible indices simple tensor contraction given applying taking contraction multiple times attach monomial map simple tensor contractions extended linearity spaces order two tensors contraction correspons matrix vector multiplication note simple order one tensor contracted vector consistent define attach constant map order zero tensor extension seen generalzation affine map zero order tensors account translation sajovic vuk applying contraction vector multiple times yields polynomial map theorem program expansion infinite tensor series point expressed multiple contractions proof show lhs dhn rhs lhs rhs functions coinciding taylor series therefore equal dhn dhn follows trivially theorem operator automorphism programming algebra stands bilinear map remark generalized shift operator operator evaluated broad generalization shift operator wiener theory presented section offers mere shift become apparent coming sections generalized shift operator denoted choice arbitrary omit expressions brevity operational calculus programming spaces remark independence operator coordinate system translates independence execution thus expression invariant point execution program fact explore section remark corollary implies enables efficient implementation operator operator program composition section forward khan barton reverse hogan mode automatic generalized arbitrary order single invariant operator theory demonstrate perform calculations operator level applied particular programming space condensing complex notions simple expressions theorem program composition composition maps expressed exp exp operator pairs maps differentiation operator applied first component second component proof show lhs dhn rhs lhs rhs functions coinciding taylor series therefore equal lim lim exp exp exp stands partitions thus lim exp taking consideration fact evaluated point evaluating expression equals bruno formula lim exp sajovic vuk theorem enables invariant implementation operator program composition expressed tensor series second map exp exp operator pullback generalized shift operator map operator exp generalized shift operator remark unified program written applying operators exp projecting onto space spanned equivalent forward mode automatic differentiation applying operators exp reverse order projecting equivalent reverse mode automatic differentiation forward reverse mode generalized arbitrary order obtainable using operator fixing appropriate map generalizes concepts single operator remark operator generalized notion pullback arbitrary operators thus descendants exponents operator grants invariance point execution program important proving algorithm correctness analogous principle general covariance see section book hanian general relativity invariance form physical laws arbitrary coordinate transformations corollary operator commutes composition proof follows theorem remark explicit evaluations corollary wise choice evaluation points operational calculus programming spaces turn towards easing calculations completing level operators derivative exp exp note important distinction operator derivative may compute derivatives arbitrary order pullback operator example operator level computation example compute second derivative exp exp equations using algebra correct applications exp exp operator always shifted evaluating point thus behavior limit importance taking limit expression obtain operator exp thus without imposing additional rules computed operator second derivative composition directly level operators result course matches equation evident example calculations using operators far simpler direct manipulations functional series similar done proof theorem enables simpler implementation functions arbitrary programming function spaces space spanned derivatives compositions may expressed solely operators using product rule derivative composition operator derivative general shift operator thus explicit knowledge rules compositions unnecessary contained structure operator exp using standard rules shown example similarly higher derivatives composition also computed operator level exp sajovic vuk order reduction nested applications useful able use derivative program part program must able treat derivative program coding original program theorem order reduction exists reduction order map following diagram commutes satisfying projection operator onto set corollary differentiable derivative theorem derivatives program extracted gained ability writing program acting derivatives another program stressed crucial lacking models pearlmutter siskind usage reduction order map constructs section demonstrated section functional transformations programs let suppose hardware optimized set functions set manufacturer technological advances switching hardware common lead decline performance thus would like employ transformations program basis common settle suboptimal algorithm hardware algorithm depends set whether spans classic example transformation basis fourier transform using developed tools problem solvable using linear algebra let denote projection operator onto basis vectors theorem map space programs space polynomials unknowns constructed using operator let denote basis space polynomials interpret vector linear combinations one choice would monomial basis consisting elements span span operational calculus programming spaces tensor basis transformation tensor basis transformation inverse set consequentially hardware upon set conditioned tensor computed used transforming arbitrary programs using operator coordinates program basis expression represents coordinates program basis thus program expressible linear combination components span used projection operator expression still represents best possible approximation original program components basis remark makes sense expand set mutual nested compositions gain mappings computing tensor increase power method special case functions describe special case subspace space functions useful set contains functions common basic operations programming language change one single real valued variable time case value changed variable described function location value saved given standard basis vector map given tensor product start construction programming spaces programming space functions instead maps analog theorem corollary easy verify since tensoring elements commutes operator analytic virtual machine terms functions enabling implementation operators functional transformation becomes much since set functions generated functions form sajovic vuk theorem suppose subspace functions suppose basis space polynomial maps basis polynomial functions matrix block diagonal block along diagonal corollary tensor basis transformation block diagonal theorem tensor basis transformation found simply inverting block note however special case model basic operations change several memory locations general model presented paper also note main goal work develop methods analysis computer programs making programming spaces maps much appropriate programming spaces functions control structures restricted operations change memories content along side assignment statements know control statements statements control statements directly values variables change execution tree program interpret control structures map spline control structure divides space parameters domains execution program always entire program divides space possible parameters set domains programs execution always program may general pnk operator point program course dependent initial parameters also expressed piecewise inside domains int int pnk int theorem program containing control structures domain int operational calculus programming spaces branching figure transformation diagram proof interior domain open entire domain int union open sets therefore open thus evaluations computed open set removing boundaries problems might otherwise occurred theorem follows directly proof theorem argument branching programs domains done conditional statements conditional causes branching programs execution tree proposition cardinality set domains equals number branching point within program remark iterators change exit conditions within body cause branching section concerns employing derived theorems propose linear treatment branchings avoid exponential threat proposition applications theory theorem program equivalently represented applications operator analytic programs number branching points within program proof source code program represented directed graph shown figure branching causes split execution tree completion returns sajovic vuk splitting point theorem branches viewed program holds theorem thus source code contains branches branching counting branch leading upon application operator needed total theorem branches analytic remark images operator elements original space may composed thus following makes sense holds true permutations applications operators visible figure remark practice always use projections operator finite order resulting therefore must take note following relation holds min composing two images applied operator projected different subspaces transformation tensor needed computed applied program running said hardware holds true branch theorem freely composed amongst remark images operator elements consisting maps evaluation composition tailor made efficient implementation methods parallelism abdelfattah computable complexities generalized tensor networks let element virtual memory analytic machine element seen map definition general tensor network sequence maps called layers defined recursively equation operational calculus programming spaces function weights bias activation may look general tensor network layers map case mean remark common neural network tensor network weight generalizations recurrent socher convolutional krizhevsky deep residual neural networks mechanisms long short term memory hochreiter schmidhuber easily generated model elements programming space general tensor network represented deeper common neural network equivalence means general tensor network fewer layers provide results deeper common neural network mitigating vanishing gradient problem described pascanu occurs training due depth network coupled machine precision remark existing architectures like theano theano development team tensorflow abadi others could easily adapted handle general tensor networks programs general tensor networks let procedure interest source code two branching points like shown figure theorem corollary transformation hereon denoted proposition image application operator program general tensor network activation function identity map layer transformations programs general tensor networks transformed program equals original program theorem corollary practice always working virtual memory equality becomes approximation thus treat transformation original program initialization weights bias general tensor network trained motivates modernizing historic taylor series evolving generalized shift operator sajovic vuk definition neural tensor series assume program written composition general tensor network defined set activation functions weights called neural tensor series order program point activation functions final layer remark neural tensor series transforms common program trainable general tensor network naturally extending theorem transformation diagram figure example setting program transformed common neural network wide applications practice algorithm providing approximate solution available neural tensor series serves great initialization point training leading process boosting freund converting weaker learner strong one currently neural networks give best results many problems described method likely provide improvements existing methods general tensor network neural tensor series program provides elegant way expressing neural computations operational calculus direct implications study understanding concepts differentiable neural computers graves neural reed freitas interchanging neural processes programming spaces reveals new directions explore might gain insight one identifying equivalence relation computer program general tensor network function taylor series enhanced activations thus may appropriately employed purpose analysis computer science path well walked remark coefficients maps allowing efficient implementation gpu parallelism claim thus general tensor networks may employ common neural networks generalize compositions tensor networks general programs methods control structures branching presented section apply general tensor networks proposition arbitrary composed general tensor networks element differentiable programming space thus treated operational calculus operational calculus programming spaces remark layer composed arbitrary element differentiable programming space allows algorithmic coding trainable memory managers generalizing concepts long short term memory hochreiter schmidhuber easing implementation networks capable reading writing external memory graves freeing semantics design process training general tensor networks transformed programs elements programming space operational calculus operators freely applied corollary thus corollary derivatives extracted reduction order map theorem derivatives order extracted projecting components used well established training methods industry proposition using operator exp theorem forward reverse mode automatic differentiation generalized arbitrary order implemented general tensor networks remark operational calculus applied training process general tensor network element differentiable programming space thus training process studied bengio analyzed trained thornton adapted particulars problem solved proposition training methods enabled operational calculus presented paper apply compositions general tensor networks arbitrary programs allows seamless trainable transitions code formulations naturally extending transformation diagram figure analysis operational calculus new approaches program analysis object study section demonstrate intertwine algorithmic control operational calculus algebra study properties denote fact object property suppose desire procedure way property changing element little possible sajovic vuk usually procedures construct may even explicitly exist easier task construct procedure whose output allows deduction whether property propose algorithm transforming procedure testing property procedure imposing property onto object domain employ operational calculus probe procedures inner structure explore interacts elements domain activity profiles property measures able determine conceptual steps important result testing procedure somehow measure activity step simple example measure activity could simply norm appropriate derivative measures rate change let assume part called conceptual step procedure definition activity profile function called measure activity conceptual step value activity level given element domain taken input vector represents activity profile procedure definition property measure function called property measure measuring amount property object exists procedure could also specific containing control structure ilustrated figure operator considers variable parameters variables operational calculus programming spaces algorithm construct property measure procedure construct property measure extract aik end end initialize set generate property measure mxj aik add mxj end return end procedure activity property measures generated starting element without property use established optimization technique optimize increase measure mxj increase measure mxj results new object possessing property theorem appropriate choice activity profile algorithm transforms procedure testing property procedure increases property measure object domain corollary theorem existence procedure testing object validity sufficient constructing procedure transforming object valid one assumption increase property measure sufficient serves simulation modeling object study procedure opens new doors analyzing phenomena may observe evolves iterations study stages change serves useful insight designing procedures causing change phenomena sajovic vuk algorithm appoint property procedure appoint property initialize path step txj extract compute energy mxj extract derivative add end update step insert end return end procedure example derived procedures given algorithm generalizations methods already present practice example demonstrates method employed take measure activity norm derivative respect variable parameters property select set txj txj highest measure activity elements property measure property simply sum squares norms derivatives selected completes algorithm corollary derivatives respect input computed reduction order map theorem using norm map property measure mxj assuming needs optimization property measure completes algorithm represents neural network stands layer network neurons variable parameters algorithm gives procedure acts operational calculus programming spaces similarly google deep dream project see mordvintsev neural algorithm artistic style gatys special cases algorithm however algorithm may applied program neural networks conclusions existence program embedded virtual reality forming system objects undergoing change virtual space reality inhabited studied science revealing principles laws virtual reality inhabited programs yet lies tougher task laws system simultaneously observed constructed universe philosophic precision programs reinforces need language capable capturing also constructing digital phenomena feat demonstrated analytic virtual machines operational calculus inspired endeavors feynman heaviside see carson physics applied operational calculus programming spaces allowing analytic conclusions algebraic means easing implementation yielded operator program composition generalizing forward reverse mode automatic arbitrary order single operator theory use algebra operational calculus demonstrated calculations manipulations performed operator level operator applied particular program language presented work condenses complex notions simple expressions enabling formulation meaningful algebraic equations solved functional transformations programs arbitrary function basis derived useful tool adapting code given hardware among formulations invariant choice programming space also point execution program introducing principle general covariance programming principle exploited designing methods transformations interchangeably applied practice section methods allow seamless transitions transformed forms original code throughout program operational calculus provides mere means calculating derivatives depth allows merging modern discoveries known old truths form neural tensor series construct useful coming terms horizon beyond theoretical truths become pragmatic approximations allows treat idealistic case deprecated initial prediction model trained widespread applicability seen process boosting already enriched science general tensor network neural tensor series program might gain insight one identifying hopefully bridging gap understanding continuous discrete computation operational calculus provides rigorous framework discussion explored section authors next subject study sajovic vuk acknowledgments first author extends gratitude jure proofreading proofs borut general guidance academia references abadi tensorflow machine learning heterogeneous systems software available abdelfattah tensor contractions gpus procedia computer science issn international conference computational science iccs june san diego california usa atilim gunes baydin automatic differentiation machine learning survey yoshua bengio optimization hyperparameters neural computation john carson heaviside operational calculus bell system technical journal issn richard feynman operator calculus applications quantum electrodynamics phys oct yoav freund short introduction boosting society artificial intelligence leon gatys alexander ecker matthias bethge neural algorithm artistic style arxiv mark girolami ben calderhead riemann manifold langevin hamiltonian monte carlo methods journal royal statistical society alex graves hybrid computing using neural network dynamic external memory nature advance online publication oct issn article kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition arxiv sepp hochreiter schmidhuber long memory neural computation robin hogan fast automatic using expression templates acm trans math july issn kamil khan paul barton vector forward mode automatic generalized derivative evaluation optimization methods software operational calculus programming spaces alex krizhevsky ilya sutskever hinton imagenet deep convolutional neural networks advances neural information processing systems pages alexander mordvintsev christopher olah mike tyka inceptionism going deeper neural networks url https hanian hans remo gravitation spacetime norton isbn razvan pascanu tomas mikolov yoshua bengio training recurrent neural networks icml barak pearlmutter siskind putting automatic back part dynamic automatic nestable fast cvs ece technical may barak pearlmutter siskind putting automatic back part wrong cvs ece technical scott reed nando freitas ternational conference learning http neural representations iclr inurl richard socher lin chris manning andrew parsing natural scenes natural language recursive neural networks theano development team theano python framework fast computation mathematical expressions arxiv may chris thornton combined selection hyperparameter optimization algorithms proceedings acm sigkdd international conference knowledge discovery data mining pages acm norbert wiener operational calculus mathematische annalen sajovic dcpp url https sajovic implementing operational calculus programming spaces computing arxiv
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packet speed cost mobile wireless networks riccardo stavros roberto ioannis electrical electronic information engineering university bologna italy department informatics athens university economics business greece department engineering university cambridge email toumpis feb department abstract mobile wireless network dtn model proposed analyzed infinitely many nodes initially placed according uniform poisson point process ppp subsequently travel independently along trajectories comprised line segments changing travel direction time instances form poisson process time selecting new travel direction arbitrary distribution nodes maintain constant speed single information packet traveling towards given direction using wireless transmissions sojourns node buffers according member broad class possible routing rules model compute averages speed packet travels towards destination rate wireless transmission cost accumulates complexity problem employ two intuitive simplifying approximations simulations verify approximation error typically small results quantify fundamental exists mobile wireless dtns packet speed packet delivery cost framework developed general versatile used starting point index terms network dtn geographic routing information propagation speed mobile wireless network parts work appear preliminary form work submitted ieee transactions information theory ntroduction networks dtns packet delivery delays often comparable time typically takes network topology change substantially means packets opportunity take advantage topology changes important class dtns mobile wireless dtns communication wireless channel changes topology due node mobility networks appear disparate settings may comprised sensors smartphones vehicles even satellites propose analyze mobile wireless dtn model consisting infinite number nodes moving infinite plane node moves constant speed along straight line choosing new travel direction given distribution time instances forming poisson process nodes move independently assumed single information packet needs travel destination located infinite distance given direction two modes travel possible wireless transmissions physical transports buffers nodes wireless transmissions instantaneous come transmission cost function vector specifying change packet location due transmission simply distance covered therefore cost may anisotropic physical transports hand associated cost introduce delays packet alternates two modes travel using routing rule selected broad class rules described terms two quantities forwarding region potential function setting define two performance metrics characterize specific routing rule used first one packet speed defined limit packet travel time goes infinity ratio distance covered divided packet travel time second one normalized packet cost defined limit packet travel time goes infinity ratio cost incurred divided distance covered generality mathematical complexity model order compute values two metrics introduce two simplifying approximations allow use tools theory markov chains approximation section approximation section approximations judiciously introduce renewals node mobility process assumptions show packet travel direction described markov chain time slot chain corresponds either packet transmission time interval packet travels along straight line buffer node general natural assumptions class routing rules considered see sections show markov chain uniformly geometrically ergodic transition kernel precisely identified compute unique invariant distribution actual values two performance metrics computed explicitly terms distribution simulations results see figs section show errors introduced approximation modest typically small namely less average numerical results present furthermore qualitative trends revealed analytical results cases confirmed simulation experiments particular demonstrated expected packet travel faster towards destination higher transmission cost due frequent use wireless transmissions rest paper organized follows section discuss related work field section iii introduce precise network model corresponding performance metrics analysis carried sections section vii present numerical results specific setting section viii contains concluding remarks finally appendix present technical proofs computations elation rior ork since gupta kumar celebrated work networks immobile nodes asymptotic analysis adapted various authors study networks mobile nodes employing wireless transmissions physical transports notably line work initiated continued among others throughput delay explored works routing protocols make use direction node traveling considered protocols examined network finite area mobile nodes move along straight lines change travel direction random times forming poisson process assumed node creates packets immobile destination whose location known nodes employ constrained relative bearing geographic routing protocol packet remains buffer node node moving effectively enough along sufficiently good direction towards destination otherwise packet transmitted suitable node whenever node available nearby scheme shown achieve throughput per node bounded delivery delay asymptotically number nodes network increases compared works pursue analysis take local view focusing cost delay forwarding single packet avoiding calculation relevant metrics multiplicative constants recently topic percolation mobile wireless networks replication single packet across network combination wireless transmissions physical transports attracted significant interest settings packet percolates finite speed except trivial case node density sufficiently high giant network component exists time instant problem computing speed studied authors consider networks wide range mobility models replication packet constrained present work study travel single packet copy towards specific destination opposed spread replication whole network significantly differentiates application work flavor analysis numerous works focused mobile wireless dtn models nodes constrained move along common fixed line models suitable vehicular dtns cars moving highways motivated part questions related road safety issues example authors consider highway comprising two lanes vehicles moving opposite westbound eastbound directions vehicles travel speed distances cars lane exponentially distributed different means two lanes study speed single packet travels eastbound direction using modes transport authors identify two distinct regions propagation packet depending specific values problem parameters either packet essentially moving speed nodes speed increases node densities general versions models studied compared works model significantly challenging moreover properly selecting distribution node travel directions results applied urban settings directions appropriately constrained although works mentioned far mainly theoretical nature number works simulation studies hybrid routing protocols employ modes transport numerous different protocols considered example moving vector move protocol favors transmissions nodes scheduled pass closest destination whereas aerorp protocol favors nodes traveling fastest towards destination see references therein examples compared works analysis gives theoretical results performance general class routing protocols tools stochastic geometry also employed studying networks node mobility crucial performance contrast prior work mentioned physical transport data example authors investigate model mobile node moves along straight line plane stationary base stations bss placed according poisson point process node contact two closer threshold distance setting authors show node comes contact bss according process observation used studying quality service qos experienced node streams video bss computing distribution download times files downloaded node bss authors study wireless sensor network comprising mobile sensors distributed infinite plane sensor moves along straight line fixed random direction random speed sensing model authors compute values various performance metrics related quality target coverage provided network namely compute percentage area covered given time instant well time needed target located outside coverage region sensed mobile immobile targets finally authors study network nodes moving according independent brownian motions two nodes contact whenever within threshold distance authors study three important random quantities time target mobile immobile comes contact nodes time nodes come contact points given subset time target comes contact node belonging giant network component although works involve physical transport information tools develop work may applicable many scenarios consider example incidence rates derived section used mobile sensor setting compute rate mobile sensor arbitrary sensing region encounters targets already mentioned mathematical complexity network model generality routing protocols consider necessitated use approximations alternative approach avoid approximations altogether arriving exact results starting much simpler network model approach taken network studied network consists locations arranged ring two mobile nodes perform independent random walks single packet travels clockwise direction buffer one nodes gets transmitted one node collocated current packet carrier moving direction node traveling clockwise direction using probabilistic tools theory markov chains explicit expressions derived average packet speed average number wireless transmissions per time slot finally note elements analysis hand first appeared compared work hand work notably differs follows regarding network model used nodes change travel direction restrictive class routing rules used regarding developed analysis packet trajectories modeled using process opposed markov chain approximation cruder approximation used elaborate difference two approximations section iii etwork odel node mobility time infinitely many nodes placed plane according uniform poisson point process ppp node density subsequently node travels independently rest nodes according following random waypoint mobility model rest work travel directions specified terms angle form positive direction node selects random travel direction according necessarily uniform direction density node moves direction along straight line constant node speed random duration time follows exponential distribution mean parameter node turning rate node picks another random travel direction density travels direction another exponentially distributed amount time mean infinitum random variables rvs independent density used describe situations nodes preferred travel directions example city center expect nodes traveling along two main axes results hold require take values set bounded away however order keep exposition simple rest discussion also assume strictly positive everywhere indeed zero subset set removed consideration subsequent analysis applies without change note time instants travel direction given node changes form poisson process rate displacement theorem theorem time instant locations nodes follow ppp density also note distribution duration time node keeps travel direction function current direction travel direction given node fixed time density transceiver model nodes equipped transceivers exchange packets suppose node wants transmit packet node whose relative location respect node described vector node distance node direction assume transmission wireless transmission cost fixed cost function also assume packets length transmissions instantaneous remarks transceiver model choice order first cost used model energy dissipated transmitter order packet reach relative location cost lost throughput silence transmitters transmission received correctly receiver second allowing cost function vector length allows treat cases anisotropy environment example environment expect energy dissipated transmitting given distance directions less energy directions signals former case pass many buildings third interpreted expected value transmission cost case random due fading results appropriately interpreted continue hold case provided sources randomness cost independent sources randomness make reference interpretations rest work fourth assumption transmission instantaneous made mathematical convenience reasonable context indeed interested measuring delays comparable time needed topology change significantly whereas time needed transmission packet typically locations transmitter receiver nodes vicinity change perceptibly finally model explicitly capture interaction packets contention channel need packets share available bandwidth implicitly modeled use wireless transmission cost function traffic model consider single tagged packet created time must travel destination placed infinite distance away packet source loss generality take destination direction positive assumption packet destination located infinite distance away made mathematical convenience plan calculate performance metrics using invariant distribution markov chain reason necessary length packet travel infinite expect metrics relevant design real networks provided packets travel finite small distances given destination packet direction positive following define travel direction better travel direction therefore packet changes travel direction better one given nodes travel speed starts approaching destination faster also use terms equal best worse worst travel directions sense packet travel destination using hybrid routing rule uses combinations wireless transmissions geographic part sojourns along buffers nodes part stages irrespective used always break travel tagged packet towards destination infinite sequence stages stage corresponding either single wireless transmission case call wireless transmission stage transmitter receiver stage single sojourn buffer node carrier travel direction change call buffering stage observe new stage occur even carrier changes direction packet stays therefore stage associated exactly one linear segments comprising packet trajectory since nodes change directions exponential times packet destination located infinite distance away source infinite number stages probability stage associate number rvs firstly let denote carrier travel direction case buffering stages travel direction receiver case transmission stages let time instants stage starts ends respectively duration observe transmission stages buffering stages let changes coordinates packet due wireless transmission stage let associated transmission cost buffering stage ywi likewise let change packet due buffering stage stage transmission stage observe cos finally write refer change progress collect rvs table table associated stage quantity stage index carrier buffering stage receiver transmission stage travel direction stage time instant stage starts time instant stage ends stage duration progress due transmission stage change due transmission stage cost transmission stage progress due buffering stage progress stage symbol cos performance metrics describe performance employed terms packet speed defined lim lim normalized packet cost lim sequel show appropriate conditions limits indeed exist constant probability packet speed represents limit average rate units distance time packet makes progress towards destination number stages goes infinity similarly represents limit average rate units cost distance cost accumulated long run packet progresses towards destination although straightforward estimate values simulation hard determine analytically one reason sequence form markov chain therefore one would consider complete chain state space describing positions travel directions nodes plane given time clearly daunting task reason introduce two approximation assumptions create artificial regeneration epochs analysis assumptions chosen judicious manner allowing apply tools markov chains guarantee induced approximation errors computations modest size indeed shown case numerous simulation examples wide range parameters finally expect exist cost speed efficiently designed makes heavy use wireless transmissions expect packet travel fast towards destination significant cost hand efficiently designed makes light use transmissions cost low packet also make slow progress towards destination simulation results also verify existence class rrs considered paper routing rule rest work limit attention following class rrs described terms forwarding region potential function first need introduce simple notational convention notation node locations described polar coordinates always understood relative locations one node relative another relative origin slight abuse notation perform operations locations written euclidean coordinates example locations nodes respect origin respectively location relative let forwarding region nonempty closed bounded convex subset defined terms arbitrary bounded boundary function observe also assume throughout cost function bounded bounded region arbitrary node located translated suppose packet node origin suitability node within either current holder another one located position traveling direction described potential function higher potential suitable node different choices two functions give rise different rrs within class make following assumptions assumption continuous strictly monotonic function following sense assumption also belong assumption says node changes travel direction strictly better one becomes strictly appealing buffering packet clearly reasonable choice potential excluding case equality simplifies analysis allows conclude time instant nodes different potentials probability allowing equality would require longer substantially different analysis performance protocols using potential functions equality may hold approximated well slightly modifying potential adding small corrective term therefore assumption limit significantly scope work assumption means node located traveling direction less appealing node located traveling direction according node located origin node also less appealing node forwarding region words nodes agree among times two nodes better buffering packet otherwise may routing loops clearly geographic routing context reasonable choice potential function naturally satisfy assumption two assumptions introduced later analysis collectively four assumptions satisfied many perhaps reasonable choices functions adequately covering spectrum routing protocol design requirements assumptions made partly mathematical convenience could relaxed various different directions without making analysis substantially harder stress analysis require specification particular choices functions particular consider specific example section vii present numerical results defined key concepts specify routing rule routing rule packet travels buffer carrier node another node refer eligible node found eligible lies potential greater nodes within packet instantaneously transmitted rule applied either another eligible node immediately found case packet transmitted time instant sojourn buffer node initiated table collect quantities used far modeling network table uantities notation used network model specified ection iii quantity node density direction density node speed node turning rate transmission cost forwarding region boundary function potential symbol approximation consequences introduce first two approximations pertains happens stages approximation moment receiver receives packet transmitter complete mobility process reinitialized except position travel direction node maintained nodes appear whose potential greater removed moment node carrying packet changes travel direction mobility process except maintains position travel direction created nodes within whose potential greater max removed note mean nodes placed time intuitively approximation introduces regeneration points mobility process markov chain amenable analysis may later defined however without eligible nodes unexpectedly appearing nowhere due ineligible nodes appear however nodes might already present effect reshuffling performance significantly affected call approximation differentiate approximation used termed approximation whenever node receives packet changes travel direction mobility process regenerated keeping node position travel direction without removing nodes even coarser basic assumption whenever node receives packet mobility process keeping node position travel direction also without removing nodes note derivations based alternative approximations notably simpler information lost setting trajectory tagged packet modeled random process simpler eventually develop section ransmission tage nalysis first step analysis section compute explicit expressions number quantities related follows wireless transmission stage setting shown fig follows node traveling direction received packet node position relative quantities interest functions write locations fig setting section eligible node within however always case let expected number nodes whose potential greater infinitesimal area element corresponding indicator function equal condition holds also let denote probability event contain eligible node new buffering stage commence moment node receives packet event occur nodes whose potential greater potential number nodes poisson distribution mean therefore exp finally let joint density location travel direction eligible node packet immediately transmitted see fig equal zero node exist given choice order obtain useful expression first observe node least suitable node keeping packet also intersection frs eligible node may found approximation joint density location direction node receive packet node better expected number nodes derivation distributed infinitesimal area element corresponding number poisson exp combining cases exp observe must due fact upon reception packet either sojourn start another transmission take place probability uffering tage nalysis second step analysis section compute explicit expressions number quantities related follows buffering stage specifically suppose time buffering stage starts packet buffer node traveling direction buffering ends time partition event corresponding end buffering stage four families disjoint events one describing different manner buffering end use second approximation introduced section compute probability events four families events given value first define collection events event buffering ends time node changes travel direction eligible node found second let event buffering ends time node changes travel direction eligible node immediately found location traveling direction third collection events consider event buffering ends time still traveling direction node located changes direction previous thus becoming eligible define fourth family first need introduce another mild assumption potential complementing assumptions section regarding let denote subset node traveling direction fig therefore nodes enter outside immediately become eligible assumption assume region convex let threshold curve parametrized curve separating assume curvature uniformly bounded differentiable respect almost every exists finite constant independent magnitude derivative respect bounded almost note assumption implies length curve bounded property obviously satisfied reasonable choice potential provided parametrization suitably chosen concreteness also mention derivative respect taken coordinates let denote unit vector perpendicular curve location specified pointing direction lower potential observe changing traces infinitesimal line segment length perpendicular see fig clearly node hits curve outside immediately becomes eligible define last collection events interest denotes event buffering ends time eligible node appears position boundary corresponding traveling direction finally write note four cases cover every possible scenario probability fig setting used defining family transition rates let arbitrary let denote infinitesimal area element location let slight abuse notation define transition rates dsdt intuitively rates describe infinitesimal probability buffering stage end exactly one four possible scenarios discussed infinitesimal duration time formally could define via limit similarly three transition rates proceed derive expressions terms network model parameters specified earlier table slight abuse terminology notation subsequent discussion omit adjective infinitesimal time referring quantities sides simply regarding probability side equal product five different quantities probability node change direction interval probability pick direction probability eligible nodes potential greater recall derivation exp probability say event occur probability say event occur observe bounded probability node region area change travel direction time interval duration claim bounded bound length curves specified assumption indeed expected number nodes given travel direction density cross whose length less time interval relative speed less less integrating follows expected number nodes less distribution total number poisson probability node cross curve time interval greater note similar arguments used calculation three transition rates show probability event different type occurs affect rate arguments straightforward omitted combining estimates ignoring terms order follows exp regarding note probability side zero condition implies eligible node changed direction however probability expressed product four different terms probability node change travel direction interval probability new direction lead lower potential otherwise packet would stayed probability node specified location specified travel direction probability node better node derivation exp therefore exp regarding probability side zero otherwise equal probability node within specified area node turn direction therefore multiplied probability regarding rate observe nodes move direction appear node moving relative speed fig also observe order probability side nonzero inner product must negative shown figure nodes travel direction hitting boundary outside probability equal density nodes traveling direction multiplied area parallelogram appears shaded figure sides length angle therefore sin noting inner product cos sin obtain max fig setting used calculating transition rates computed expressions finally define one last family transition rates used subsequent derivations first given value define family events event buffering ends time eligible node appears position curve traveling direction also define transition rates observe simply different representation easily recovered knowing indeed fixing rate specifies transition rates eligible node arrivals locations provided function hand rate already contains information particular numerical calculations later calculate rate specific pair follows first discretize defining values cover associated interval length intervals partitioning discretize defining values associated area areas partitioning map location nearest denote resulting map setting discretized version aggregate rates also define aggregate rates follows interpretation first four rates one multiplied infinitesimal conditional probability event corresponding family occur time given packet traveling direction last one rate multiplied gives probability buffering stage end time given observe must union events belonging families event node changes direction happens rate therefore approximation consequences transition rates events four families defined section independent duration buffering stage intuitively time progresses memory accumulates probability occurring changes fact significantly complicates analysis required computing probability specific one events occurs adopt following simplifying assumption approximation incremental event time intuitively approximation probability buffering end specific manner change stage progresses equal probability happen right moment buffering starts mobility process restarted due approximation particular integrating expression implies memoryless therefore approximation exponentially distributed rate furthermore approximation makes possible obtain versions expressions rates example adopting slight abuse notation event exp third equality follows approximation last equality definition fact conditional exponential rate integrating obtain exp working manner families arrive similar results summarizing erformance etrics section derive expressions average packet speed cost induced network model expressed terms invariant distribution appropriately defined markov chain following last technical assumption need impose potential function assumption value equal constant coupled monotonicity assumption simply states direction uniformly worst irrespective location candidate neighbor note however behavior function strongly depend good locations favored terms potential assigned long nodes locations traveling direction therefore assumption clearly significantly restrictive technical terms used establish irreducibility chain defined become evident analysis assumption could relaxed cost significantly complicating arguments involved pursue direction markov chain define state associated stage wireless transmission stage buffering stage associated state space takes values transmission state space buffering state space assumptions node traveling direction never receive packet node irrespective location node traveling direction therefore pairs included observe due approximation process forms markov chain buffering stage start stage complete mobility model restarted except carrier kept travel direction contain nodes potential higher likewise stage packet transmitted node node located moment received packet whole mobility model restarted except kept travel direction nodes potential higher expunged cases complete information remaining network captured current state distribution chain may described follows assume arbitrary initial state given chain moves state according following family conditional distributions derived previous section conditional density kbb conditional density kbw nonzero conditional density function given finally conditional density sequel refer kbb kbw kernel functions since used fully specify transition kernel chain ergodicity section establish approximation approximation assumptions markov chain ergodic unique invariant distribution converges geometric rate let denote lebesgue measure denote lebesgue measure point mass point write measure defined state space equipped usual borel first result describes behavior chain consequences stated detail see relevant background markov chains theorem proved appendix theorem approximation approximation assumptions markov chain aperiodic uniformly ergodic state space unique invariant measure converges uniformly geometrically fast particular constants initial state measurable set measurable function probability one initial state important ingredient proof theorem following domination condition verified appendix intuitively lemma says irrespective current state probability least chain uniformly distributed buffering state two time steps lemma doeblin condition let denote measure measurable another ingredient proof part theorem provided following reachability bound lemma proved appendix lemma let denote measure measurable main implications theorem results stated following corollary proved appendix order state need additional definitions given arbitrary state let exponentially distributed rate otherwise similarly given let distribution given defines new markov chain state space suppose distribution let defined conditional write induced joint distribution corollary initial state following ergodic theorems hold probability one lim lim lim cos lim particular final step analysis provide expressions performance metrics defined section following results stated without proof immediate consequences corollary corollary initial state limits defining performance metrics respectively exist probability one given cos cos details regarding numerical computation expectations given section appendix vii umerical esults section compare approximate results performance metrics obtained previous sections corresponding simulation results specific choices potential function setting consider potential function packet constantly tries find nodes good travel direction regardless relative location provided course within consider specified boundary function cos ellipse whose major axis length along left focus origin eccentricity boundary function drawn fig different choices parameters note larger values make routing protocol aggressive finding nodes send packet whereas larger values make routing protocol selective regarding relative locations neighboring nodes important special case circle radius routing protocol gives packet node direction better current holder long two nodes within distance choice boundary function may possible nodes know travel directions relative locations exchange packets whenever within communication radius fig left boundary function two different values parameter three different values parameter right density three different values parameter finally assume transmission cost quadratic density travel direction take elsewhere therefore positive constant four intervals centered directions positive negative whereas outside ranges zero density directions extreme small values model situations uniform density nodes move along direction one two perpendicular axes would happen example vehicular network nodes moving rectangular road grid fig also plot density three different values parameter table iii collected quantities used calculations section along default values values used computations unless explicitly stated otherwise table iii uantities default values used ection vii quantity node density direction density node speed node turning rate transmission cost boundary function potential symbol elsewhere cos default value uniform results fig shows effects shape eccentricity length vary packet speed packet cost subsequent figures results obtained earlier analysis shown solid black lines corresponding simulation results shown dotted red lines fig versus solid black lines depict analytical results dotted red lines depict simulation results observe length increases packet speed increases packet cost exemplifies fundamental two metrics increase speed gets larger becomes likely node good travel direction available carrier changes direction bad one also node farther ahead average reason also transmission cost function quadratic also increases increases fact figure suggests expect speed diverges infinity increases since expected progress per wireless transmission increases hand cost diverges infinity increases even regarding effects eccentricity observe starting increasing initially leads higher speed lower cost natural value corresponds circular therefore neighboring nodes whose relative position towards positive given preference inefficiency rectified initially increases however increasing past actually leads increase cost indeed elliptical often happens packet transmitted nodes far away current carrier albeit excellent relative position although nodes much closer carrier relative position almost good cost quadratic inevitably increases packet cost large values hurt speed increases area reduced exact formula packet spends time traveling towards relatively bad directions buffers nodes fig plot values versus two node parameters namely node density node turning rate fig versus solid black lines depict analytical results dotted red lines depict simulation results regarding effects first observe small long node density small packet speed almost equal node speed indeed packet stays node direction significant amount time infrequent cases node changes direction another one found within relatively short time consequently packet cost also small hand small larger frequently node changes direction frequent transmissions nodes better directions hence packet cost speed get larger effect speed crucially depends fact transmissions average towards direction positive since figure use default value regarding effects fixed low density leads low packet speed packet spends extended periods time traveling towards bad directions hand large node density means packet travels fast due frequent transmissions however effect diminishes node travel direction guaranteed exist within whenever current carrier changes travel direction therefore increasing density effect hand changes understand compare regime regime first case travel packet consists wireless transmissions physical transports right direction second case consists wireless transmissions transports right direction also involves extended periods transports random directions average produce progress two cases differ significantly performance terms progress per unit time speed terms cost per unit distance positive direction transports random directions approximately zero net effect finally fig plot versus angular width size circular disk fig versus solid black lines depict analytical results dotted red lines depict simulation results observe fig increasing increases speed cost indeed nodes higher probability packet changes travel direction another node good travel direction available however contrast fig increases effects speed tend diminish indeed value probability eligible node good direction exists invariably close unity eccentricity wireless transmissions zero net effect speed packet change effects much less pronounced change little justified observing changing make directions nodes travel overall better differently distributed still effects performance metrics remarkably small finally note discrepancy simulation results analytical results generally small almost always modest one exception setting fig cases small values large values discrepancy significant however discrepancy due inaccuracy two simplifying approximations rather due accumulating numerical errors specifically regime errors due discretizations used large probability two nodes exact discretized travel direction hence potential packet arrives new node current holder changes direction note analysis assumes probability event zero excepting case discrepancy simulations analysis remains modest although increase fig indeed increases approximation invoked frequently estimated rates encounters eligible nodes deviate actual ones viii onclusions work first introduced mobile wireless dtn model nodes move infinite plane according random waypoint mobility model packet must travel destination located infinite distance away according routing rule using wireless transmissions physical transports buffers nodes routing rule defined terms forwarding region potential function specifying leads different versions routing rule model quite general notably including cases transmission cost depends direction transmission arbitrary distributions direction node travel large variety routing rules setting defined two performance metrics speed packet travels destination rate transmission cost accumulated computed performance metrics adopting two simplifying approximations approximations ensure simpler markov chain embedded system description analyzed using general tools markov chain theory assumptions intuitive furthermore shown introduce modest errors order examples considered numerical evaluations present results help quantify important exists mobile wireless dtns speed packets travel destinations rate transmission cost accumulated also methodology developed may extended variety directions include case velocity magnitude constant duration time node spends given travel direction depends velocity vector alternatively present development may also used starting point accurate analytical approaches maintaining memory stage markov chain related work much simpler settings suggests dispensing approximations altogether might formidable task interesting potential application work towards studying performance geographic routing protocols consider example simple circular radius probability going eligible node whenever packet arrives new node therefore increases physical transports become less less frequent taking limit cost readily gives performance geographic routing protocol regarding future work present setting naturally leads problem finding best routing rules achieve pareto optimal combinations delays costs tackling problem tools genetic algorithms bandit theory jointly tools stochastic geometry might fruitful strategy also assumption nodes travel speed crucial could relaxed example could use general model node travels independently others speed constant times node changes direction speed changes direction changes speeds associated consecutive trajectory segments independent random variables following given distribution likewise could assume distribution duration time node spends given travel direction depend direction hand two aspects model namely independence node trajectories changes node directions according poisson process easily relaxed would introduce new sources memory making difficult develop accurate tractable markov chain model packet trajectory ppendix first section appendix give proofs theoretical results section sections provide details numerical evaluation performance metrics intermediate results proofs establish theorem lemmas corollary several parts proofs need invoke technical quite standard arguments details omitted proof lemma obvious suffices establish result lemma events form uniqueness extension since collection finite unions intervals forms algebra generates borel suffices consider closed intervals see details rest proof restrict attention events form also note expressions derived section simple obtain following bounds transition rates exp note lower bound defined section form using lower bound upper bound fixed constant using markov property applying twice similarly form markov property exp recalling expression clearly exp combining yields required result min exp proof lemma since positive lebesgue measure find rectangle form nonempty interior idea main argument show range angles current packet holder travels direction strictly nonzero probability ineligible nodes become eligible changing travel direction better one within range since continuous image closed interval since nonempty interior must assumption also assumptions noting order must next pick min let angles guaranteed exist intermediate value theorem also observe continuous compact set uniformly continuous implies take arbitrary bound given first note therefore also therefore second inequality follows fact assumed bounded section also recall bounded ready prove inequality interval chosen last integral strictly positive nonzero bounded away zero bounded proof theorem first establish aperiodicity chain fact show state measure absolutely continuous respect measure end choose fix arbitrary state arbitrary measurable subset either first case lemma implies together markov property implies second case combining lemma lemma applied markov property obtain positivity last integral follows lemma finally using markov property required aperiodicity together doeblin bound lemma imply chain uniformly ergodic specifically lemma implies state space small drift condition holds lyapunov function theorem implies chain unique invariant probability measure distribution converges uniformly stated part theorem particular chain harris recurrent theorem implies strong law large numbers holds functions stated part theorem proof corollary since bounded hence functions ywi first two results immediately follow theorem next two let denote measure arguing proof theorem easy show new chain aperiodic also uniformly ergodic theorem implies strong law large numbers holds recalling cos last two statements corollary follow soon establish indeed since given exponential rate completing proof invariant distribution expectations let density think infinitesimal proportion time similarly let denote joint density xwi ywi think infinitesimal proportion time infinitesimal area element centered order compute functions derive balance equations follows first note proportion state transitions equal proportion transitions observe proportion transitions equal proportion rar transitions states inside likewise proportion transitions infinitesimal area element centered size proportion enters therefore integrating relevant state transitions second observe proportion transitions equal proportion transitions set compute observe state proportion time equal instances proportion transitions equal likewise proportion transitions proportion enters integrating relevant state transitions obtain order compute discretize arguments converts balance equations large linear system also use fact sum integrals equal unity see section details readily derive following expectations respect invariant distribution chain cos inner variable integration first integral numerical computation integrals number instances work need compute values multivariate function given multiple integral good example following form show calculate integral similar integrals work calculated using method mutatis mutandis first discretize variable considering values positive integer second discretize placing points uniformly inside achieved follows create points positive integer parameter region assumed bounded lies entirely within square keep points within region denoting polar cartesian coordinates respectively restrict calculating equation becomes integral calculated approximately follows associate point rectangle volume rectangles approximately partition set integral taken finally set clearly larger closer approximation true values speeding computation part computations required numerical results repeated calculation using indeed function two variables finding values requires evaluation multiple integral however assuming potential function function makes computation simpler indeed case integrals computed beforehand necessary values either numerically analytically readily available calculation starts furthermore even simpler expression speeding computation another part computations calculation using function four variables values requires evaluation multiple integral calculation significantly simplified take account special structure potential function example function simplifies given speeding computation one final computational bottleneck calculation using function three variables calculating values involves evaluation multiple integral considering case potential function first integral becomes furthermore multiple integral becomes area order speed relevant calculations computed beginning made available calculation starts numerical solution finally describe numerical solution system balance equations derived section appendix discussed restrict computing discrete sets values given respectively become perform piecewise constant approximation integrands rectangular sets constant values values integrands centers taking integral union rectangles get multiplying first equation second equation defining slight abuse notation likewise functions appearing balance equations obtain system linear system equations may interpreted expressing balance equations discrete markov chain states ensure probabilities transitions state indeed add unity cases solve system write form vector length consisting followed columns since resulting matrix stochastic construction find eigenvector corresponding top eigenvalue normalization provides required solution eferences cavallari verdone toumpis analysis mobile wireless dtns random waypoint mobility proc wiopt tempe may cavallari toumpis verdone analysis hybrid routing protocols wireless mobile networks proc ieee infocom honolulu apr kontoyiannis toumpis cavallari verdone calculating packet speed cost mobile wireless network model submitted ieee isit vasilakos zhang spyropoulos delay tolerant networks protocols applications crc press gupta kumar capacity wireless networks ieee trans inf theory vol mar grossglauser tse mobility increases capacity wireless networks proc ieee infocom vol anchorage apr diggavi grossglauser tse even mobility increases capacity wireless networks ieee transactions information theory vol toumpis goldsmith large wireless networks fading mobility delay constraints proc ieee infocom hong kong china mar sharma mazumdar shroff delay capacity mobile hoc networks global perspective trans vol jacquet malik mans silva tradeoff georouting networks ieee trans inf theory vol june jacquet mans rodolakis information propagation speed mobile delay tolerant networks ieee trans inf theory vol wang message dissemination intermittently connected communication networks ieee trans wireless vol july baccelli jacquet mans rodolakis highway vehicular delay tolerant networks information propagation speed properties ieee trans inf theory vol march zarei rahmani samini connectivity analysis dynamic movement vehicular hoc networks wireless networks baccelli jacquet mans rodolakis networks radio ranges information propagation speed properties proc ieee isit istanbul turkey july lebrun chuah ghosal zhang opportunistic forwarding vehicular wireless hoc networks proc ieee vtc spring vol stockholm sweden peters jabbar sterbenz geographical routing protocol aeronautical networks proc ieee wcnc cancun mexico mar tasiopoulos tsiaras toumpis optimal achievabnle tradeoffs networks computer networks vol madadi bacelli veciana temporal variations mobile users snr applications perceived qos proc wiopt tempe may liu brass dousse nain towsley mobility improves coverage sensor networks proc acm mobihoc may peres sinclair sousi stauffer mobile geometric graphs detection coverage percolation probab theory relat fields vol cheliotis kontoyiannis loulakis toumpis exact speed transmission cost simple wireless network proc ieee isit aachen germany june sidera toumpis delay tolerant firework routing geographical routing protocol wireless delay tolerant networks eurasip journal wireless communications networking tradeoff wireless mobile networks proc wiopt tempe may baccelli blaszczyszyn stochastic geometry wireless networks vols foundations trends networking ephremides energy concerns wireless networks ieee wireless commun vol ross stochastic processes new york john wiley sons meyn tweedie markov chains stochastic stability london cambridge university press published cambridge mathematical library edition online http royden real analysis new york macmillan williams probability martingales cambridge university press kontoyiannis meyn spectral theory limit theorems geometrically ergodic markov processes ann appl vol february
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comparative studies programming languages within diverse criteria revision rana concordia university montreal quebec canada mohammad fahim concordia university montreal quebec canada sheetal concordia university montreal quebec canada jalal concordia university montreal quebec canada marinela concordia university montreal quebec canada abstract survey programming languages javascript aspectj haskell java php scala scheme bpel survey work involves comparative study ten programming languages respect following criteria secure programming practices web application development web service composition abstractions reflection aspect orientation functional programming declarative programming batch scripting prototyping study languages context mentioned criteria level support provide one keywords programming languages programming paradigms language features language design implementation introduction choosing best language would satisfy requirements given problem domain difficult task languages better suited specific applications others order select proper one specific problem domain one know features provides support requirements different languages support different paradigms provide different abstractions different levels expressive power better suited express algorithms others targeting users question best tool particular problem aspects like security language safety prototyping capabilities language support building distributed systems support automating existing processes portability also important issues consider choosing programming language analysis discussed suitability selected languages specified criteria primarily focused primarily focused primarily focused primarily focused primarily focused programming languages javascript programming languages programming languages java programming languages scala programming languages bpel comparative studies programming languages within diverse criteria team term report related work work major influence paper written approach taken compare programming languages author emphasises difference design implementation languages decisions one influences author performs comparative study model transformation languages context mde author compares different web service technologies paradigms evolution author discusses technologies architectures context distributed systems author studies fundamental concepts programming languages study language concepts varying degree abstraction performed context several studies performed previously among several programming languages several programming paradigms observation aspectj yet compared way previous research work found great research work like really interesting motivated extensive research finally provided guidance necessary perform study overview rest paper organized follows first provide overview languages studied beginning section section section performed detailed analysis studied languages context specified criteria section consolidated results table supported brief description section conclude work incorporated statistics programming languages language overview different platform windows linux unix different instruction set executable file produced compiling code platform run platform windowsexecutable file comes code run windows compatible environment javascript language overview java compiler side converts source code intermediate code platform specific java virtual machine run intermediate code produced platform javascript language able use java objects call public methods facility earn javascript use exist java exist javascript javascript code compiled interpreters implemented java language interpreters external interface used communicate external component script code runs user side used control behavior end user aspectj language overview aspectj general purpose programming language simple practical extension java aspectj extends java language keywords writing aspects pointcuts advice code intertype declarations gregor kiczales team created new programming paradigm aop palo alto research center parc also developed comparative studies programming languages within diverse criteria team term report leading tool programming using possible create clean modular implementations crosscutting concerns tracing login user session management synchronization consistency checking protocol management etc creating new constructs aspectj provides support modular implementation range crosscutting concerns dynamic join point model join points points program advice code executed pointcuts collections join points advice special constructs attached pointcuts aspects modular units crosscutting implementation comprising pointcuts advice ordinary java member declarations language overview programming language goal language increased programmer productivity language becoming popular perfect balances simplicity expressiveness performance language developed microsoft corporation within initiative chief architect language since first version anders hejlsberg also original author turbo pascal chief architect delphi rich implementation paradigm includes data abstraction encapsulation inheritance polymorphism typically used writing code runs windows platforms although microsoft standardized language clr ecma total amount resources support platforms windows relatively small purpose common language runtime clr provide platform application development execution including functions exception handling garbage collection security interoperability haskell language overview haskell functional programming language haskell named haskell curry one pioneers lambda calculus mathematical theory functions inspiration designers number functional languages haskell functions lazy evaluation light syntax data type side effects makes possible safely abstract part function replace parameter paradigm treats computation mathematical functions avoids state mutable data strongly typed eliminating huge class easy make errors compile time possibility core dumps example java one write function accepts two arguments possible type however haskell goes allowing function accept two arguments type long type java language overview java programming language originally developed james gosling sun microsystems strongly typed object oriented provides excellent means modularizing structuring programs also supports concurrent execution multiple threads well key synchronization mechanisms integrated exception handling rich set integer data types package supports user documentation general coding style guidelines supports abstraction information hiding code portability comparative studies programming languages within diverse criteria team term report php language overview php powerful scripting language run command line computer php installed php originally created rasmus lerdorf stood personal home page released free open source project php renamed php hypertext preprocessor php especially creating dynamic web pages connectivity various database systems mysql widely used php provides native support database free project php runs different platforms compatible almost servers used today php easy learn runs efficiently server side scala language overview scala stands scala general purpose programming language designed express common programming patterns concise elegant way smoothly integrates features functional languages enabling java programmers productive code sizes typically reduced factor two three compared equivalent java application scala runs standard java platform interoperates seamlessly java libraries scheme language overview general characteristics scheme programming language supporting functional procedural meta web applications scripting application development small language core powerful tools allow language extended goal stated scheme facto standard provide base compatible implementations built typical characteristic dialects lisp including scheme homoiconic list scheme interpreted data structure source listing compiled executed another characteristic targeted scheme facto standard although language dynamically type type checking binding variable names object types done run time supports implicit polymorphism variables types bound value type scheme type safe type system typing rules type equivalence type compatibility strictly enforced implementation application function objects types support function allowed scheme language considered strict functions always strict function undefined arguments undefined due applicative order evaluation arguments passed functions scheme first lisp dialect support lexical static scoping binding variable name usage declaration inferred program text without need trace execution history dynamic scoping determine name declaration binding makes programs easier write understand prevents introduction many difficult track programming errors another characteristic scheme expressive language algorithms represented concisely scheme homoiconic feature allows express uniformly design implementation semantics language addition data handled programs expressed language comparative studies programming languages within diverse criteria team term report architecture system scheme program implementing expanded created runtime compiling interpreting newly created source code scheme reach reflective system support also supports applicative eager order evaluation arguments functions normal lazy order evaluation special forms cond special form scheme uses cambridge polish parenthesised notation expressions first element inside parenthesis function macro special form rest parameters anonymous functions defined using lambda special form functions control flow typical function call scenario implementation restores referencing environment conforming lexical scoping rules time arguments referencing environment bound formal parameters function expression inside function sequentially evaluated value last expression return value function another function scheme provides conditional expression special forms assignment sequencing iteration imperative constructs delay force lazy evaluation arguments special forms passed unevaluated expression types functions special forms scheme primitive language implementation derived defined terms primitives lambda special form used build derived functions build derived special forms macros special form used forms bound names using mechanism allows language expansion creation libraries scheme implementations implementations scheme languages include commercial free run different platforms examples graphical programming environment implementations follow pro edscheme bee drscheme fluxus livecoding environment schemeway scheme environment built atop eclipse pilo visual tools scheme rich list scheme implementations given bpel language overview general characteristics goal bpel language allow smooth integration heterogeneous systems different application domains enterprise application integration example typical interaction scenarios involve sequences stateful message exchanges two participants web services described wsdl support stateless uncorrelated message exchanges bpel built bridge gap requirements comparative studies programming languages within diverse criteria team term report bpel language automates business processes way orchestrating existing functionality services expose interfaces web services using wsdl standard language bpel processes web services described using wsdl achieve easy integration heterogeneous web services soap used messaging protocol protocol allows xml messages data transported platform neutral programming model independent way bpel process consists abstract executable part abstract process executable specifies external message exchanges partners contain internal details business process executable process defines external messages logic executable concepts relevant bpel processes message flow describe order external web services invoked control flow specify order activities within process execute data flow stateful interactions include content messages exchanged intermediate data used internal computations composed messages sent partners functions control flow bpel process specifies multiple service interactions partners coordinated achieve business goal well state logic necessary coordination introduces exception handling mechanism compensation mechanism failed unit work transaction computational model imperative nature constructs used specify internal logic data model used bpel defined using wsdl messages xml schema type scopes used encapsulation mechanisms lexical static nature scopes nested control flow construct scope scopes exist without control flow construct variables comply static rules inferred program text subroutine function constructs bpel module considered one function imperative sense subroutine internal concurrency afforded use flow construct iteration achieved using repeatuntil foreach constructs events handling achieved thought onmessage onevent encapsulated inside pick construct pick blocking nature waits specific event occur message arrive event handled within pick scope control released rest process conditional logic expressed using construct although previously existed switch construct part bpel version oasis standard updates variables achieved using assign construct bpel uses invoke request service delegate computation another web service since bpel web service exposes capabilities using wsdl interface invoked bpel capable receiving external requests using receive construct reply used return response client invoked process bpel implementations bpel implementations include activevos open source eclipse stp bpmn diagram editor eclipse bpel project orchestra fully open source extensible flexible bpel solution open source bpms eclipse apache ode open source bpel server netbeans enterprise pack bpel windows workflow foundation comparative studies programming languages within diverse criteria team term report analysis javascript default secure programming practices operating system written language therefore close operating system especially relates pointers memory access fact makes unsafe easy illegal access memory object internal exception handling memory boundary dealing arrays allocating memory space error prone language easily may introduce privacy leak memory management cases needs done programmer complicates debugging task susceptible buffer overflow checking input length function following code may lead stack flow see figure int main char name name name null execve name name null return fig stack function called comparative studies programming languages within diverse criteria team term report fig stack buffer overflow recommended follow guidelines programming mention following enforce standard coding style guidelines validate function inputs trace code ignore warning prompts produced code compilation use object oriented style hides properties encapsulates implementations prevent privacy leak guidelines exists javascript javascript features inherited java language java language lot build security features one security challenges javascript provide secure environment run mobile code java language originally built mitigate many programming mistakes known java feature memory management checks arrays boundary java virtual machine applies strict mechanism prevent stack overflow happening java javascript faces following threats malicious subclass may clone override random methods code operating privileged access probably flaws privileged access need given trusted entities running unprivileged code sandbox solve problem java library could replaced malicious implementation exception handling may show sensitive knowledge shows file path since many attacks need know file path dynamic sql creation along untrusted input leads injection flow code hidden script api code malicious big threat may introduced malicious code may use rmi ldap inject system javascript dynamically typed comparative studies programming languages within diverse criteria team term report javascript check arrays boundary strings arrays variable length threats exists previous threat may isolated use isolate unrelated code using security property private static final string static string packageaccess packageaccess null packageaccess guideline isolate unrelated code web application development used creating web applications using toolkit library developed web applications encapsulates many web details show details embedded web form components ordered hierarchical widget tree library provides web widget called wwebwidget deals html dom theses components piece text wtext table wtable line edit wlineedit following example print hello world quit button comparative studies programming languages within diverse criteria team term report code taken javascript javascript imperative scripting language mainly used internet applications helps computations associated html interpreted client browser also motivates client use websites javascript brings dynamic changing websites javascript makes computation done client side increases network bandwidth improve computations performance javascript cases used check validity input data entered user side help checking client side better computation server side otherwise server send notification message ask user redo data entry moving computation server side client side improves web computations reduces network bandwidth imagine social web application like face book lot updates happen second needs maintain privacy clients server responsible control every client connected moments face book servers going shutdown soon due loads considered huge javascript script helps reducing issue control client side following example javascript code declares two integers computations print result html body script var var var comparative studies programming languages within diverse criteria team term report script declares variables assigns values computations displays computation values displays value web services design composition web services use xml remote procedure protocol simple object access protocol soap transport layer send request receive response following figures shows deploy services well transport request receive response helps messaging establishes communication across different platforms picture taken comparative studies programming languages within diverse criteria team term report picture taken good example shows deploy web services invoke services following steps used deploy web service using sca service component architecture examples bellow taken declare services declare interface abstracts super class contains virtual functions code fragment taken comparative studies programming languages within diverse criteria team term report developer implements interface already declared previous step code fragment taken code fragment taken iii use wsdl generator generate wsdl porttype code fragment taken comparative studies programming languages within diverse criteria team term report following figure shows complete image connection previous steps code fragment taken client side use code call method implemented server side need case know service name port address javascript call web services client using script language javascript java api web services framework helps web service following example defines web service one operation called hello accepts one parameter type string notice tag webservice declare attached class web service interface webservice tag declare service name host address port number well port numer mentioned default port called webmethod tag used define web service operation parameters well implementation import import import webservice http public class hello webmethod public string hello webparam name name string name comparative studies programming languages within diverse criteria team term report return hello name following code example taken need deploy web service following wsdl generated wsdl tool used deploy previous declaration definitions xmlns http xmlns http xmlns http http http helloservice types schema xmlns http http element hello tns hello element helloresponse tns helloresponse complextype hello sequence element name string sequence complextype complextype helloresponse sequence element return string sequence complextype schema message hello part parameters tns hello message helloresponse part parameters tns helloresponse porttype hello operation hello input tns hello output tns helloresponse binding helloportbinding tns hello soap binding http document operation hello soap operation input soap body literal output soap body literal comparative studies programming languages within diverse criteria team term report service helloservice port helloport tns helloportbinding soap address http following code example taken following soap invocation helloservice service new helloservice hello port string response name following code example taken following invocation hello service import import import string endpoint http service service call call endpoint new qname hello string response string new object name following code example taken call web services using javascript code facilitate design web services need create complex client design need implement javascript code embedded html file run scrip remotely client side following figure javascript code shows call hello web service comparative studies programming languages within diverse criteria team term report script javascript init http http wsdl hello function tst icallid hello concordia function onmyresult resultado body init button javascript tst call add web method div service behavior url onmyresult following code example taken modifications dealing javascript order call web services leads simple flexible solution hence need aware transportation mechanism details soap used transport layer web service implementations abstraction supports imperative object oriented based programming style provides stack hap allocation emphasizes stack model provides multiple inheritance exception handling well interface definition garbage collection mechanism java therefore memory allocation done programmer need returned programmer support static dynamic binding cpp built enhance imperative properties add new features support object oriented programming cpp support classes consider bases object oriented programming example declare string cpp following class useful comparative studies programming languages within diverse criteria team term report class string private char str public string char char strlen int strlen str delete str support function overloading also supports namespaces methodology used group multiple classes namespaces well related properties also cpp supports abstraction encapsulation inheritance methods virtual functions polymorphism following example shows code taken previous example three types animals cat tiger ocelot different kind sound meow create pointer base class may point object one cat tiger ocelot call method meow run time implementation meow method determined run time dynamic binding called polymorphism see run result comparative studies programming languages within diverse criteria team term report code run taken javascript java language fully support object oriented concepts features inheritance polymorphism however javascript support inheritance virtual functions javascript run within browser support javascript interpreter javascript support simple paradigm allows programmer build complex data structure javascript support class construct allows defining methods properties well event handler thus allow abstract different kind objects javascript data types instance methods data types javascript inherited object objects javascript prototype javascript support namespaces methodology used group multiple classes namespaces well related properties javascript support static dynamic binding example dynamic binding importpackage var hashtable new hashtable javadate new java socket new java comparative studies programming languages within diverse criteria team term report javascript support function overloading void write string void write int javascript permits developer define classes variable types well javascript javascript objects see following example script var hello world class abstraction javascript create abstract definition object javascript start need declare namespace suppose want declare namespace animal following line animal declare class cat need include namespace following lines declares cat constructor function cattype catname catage cattype cattype catname catname catage catage following example shows define class prototyping properties methods getcattype function return getcatname function return getcatage function return getcatinformation function var str cat age return strdetails comparative studies programming languages within diverse criteria team term report dispose function alert destroying next step register class javascript support inheritance based prototypes concepts works template new objects mechanism helps programmer avoid declaring common tasks javascript support interface definition declaration interfaces help group common methods distributed among different class definitions javascript support reflection javascript support basic collection objects like date math arrays following example depicts deal array object html body script var mycars new array mycars mycars mycars car mycars figure declare variable javascript need use var keyword var keyword help declare type reflection reflection mechanism helps program dealing undefined object compile time program check based conditions going create specific object invoke method reflection programming context means build generic code deals undefined object mechanism facilitates remote method invocation serialization support reflection devadithya states two types reflections reflection reflection decision code need bind compile time decision depends comparative studies programming languages within diverse criteria team term report metadata availability compile time reflection depends program knows type information run time case program able take decision run time reflection behavior happens dynamically execution time support reflection due unrecognizing detailed type information complier hides information reflection limited features monitoring expression types querying type name following fragment shows depicts reflection classtype classtype obj previous code shows call configurable methods another example memberfunction getmemberfunction int myclassobj code invoke member function returns int accept argument remote method invocation kind reflection ability allow program run code remotely code could change depends remote host remote host could decide change implementation remote method also local host able change method call changing calling argument previous example different method name devadithya demonstrated reflection added language details see javascript java supports reflection part standard specifications reflection javascript context means ability program dynamically examine program structure get current object information instance class inherited class interface implements well class properties methods javascript build top java extendscript construct facilitates reflection behavior construct helps provide information objects name description properties well input parameters return type comparative studies programming languages within diverse criteria team term report javascript support reflection object construct provides program contents short long description method defined interface properties defined class following code example var new file myfile var props var property props props following example shows example object retrieves properties methods information data type properties obj new string string obj new string indexof slice indexof method info code gets list properties number code gets data type property aspects implement crosscutting concerns modular way pointcut considered major element pointcut composes single multiple joint points need used across multiple functions namespace joint point may evaluated compile time run time two types pointcuts code pointcuts name pointcuts information see manual following code describes code joint point comparative studies programming languages within diverse criteria team term report figure taken comparative studies programming languages within diverse criteria team term report previous code example figure taken manual depicts meaning name joint point code joint point join point could function attribute type variable depending kind pointcuts evaluated compile time runtime name pointcuts may compose types attributes functions variables namespaces code point cut execution point function repeated multiple places program possible mix name pointcuts code pointcuts within pointcut expression pointcut expression way search pattern uses special character wildcard used aggregate simplify describing collection classes cunfcting names share characters javascript xhtml advantage separate web form content helps improve separation concern aspect oriented language aop aop centers around separation concern approach means create main program invoke sub programs subprogram concern perform specific task concern logging authentication synchronization usually repeated used many places many programs login synchronization concerns called cross cutting aop helps improve memory usage synchronize cross cutting modification encapsulation cross cutting modular form javascript feature weave advice code destination code helps change behavior program without refractor original code advice code written separate file woven original code execution time java platform integration could source code level byte cide level javascript support aop utility example advise idiom three kinds aspect directions seen following example aspects new object function intro obj var function aspect objs var otype typeof objs typeof aspect throw invalidaspect otype aspect objs comparative studies programming languages within diverse criteria team term report else otype var aspect objs else throw invalidobject function aspect obj funcs var ftype typeof funcs typeof aspect throw invalidaspect ftype funcs array funcs var var fname funcs var old fname old throw invalidmethod fname function arguments return arguments function aspect obj funcs typeof aspect throw invalidaspect typeof funcs funcs array funcs var var fname funcs var old fname old throw invalidmethod fname function var args arguments comparative studies programming languages within diverse criteria team term report return ret array args null function func var osrc new string func var nsrc var osrc osrc osrc nsrc else nsrc osrc nsrc nsrc return nsrc function aspect obj funcs typeof aspect throw invalidaspect typeof funcs funcs array funcs var asrc aspect var var fname funcs fname throw invalidmethod var osrc original fname var fsrc osrc arguments fname function fsrc return true comparative studies programming languages within diverse criteria team term report functional programming rich library supports functional programming library provides support polymorphism using type inference big part standards library implemented functional model language functional programming style advantage direct mainplatation memory primitive support dealing pointers object based programming language support creating classes using class struct keyword support operator overloading overloading helps developer teach class object deal operator following example shows overload operator figure taken declare function function following syntax following example function add accepts two parameters return addition two parameters comparative studies programming languages within diverse criteria team term report int add int int return javascript javascript support functional programming javascript support standard functions map reduce select see following example map result select result reduce result map guard result javascript block scope function scope declare function javascript key word function used followed function name parameters function fname see following example html head title function definition script javascript hiding javascript function listitems itemlist comparative studies programming languages within diverse criteria team term report itemlist end hiding javascript body script javascript new array sunday monday tuesday wednesday thursday friday saturday listitems days another example function return value function average var items var sum items sum return following call function average average declarative programming imperative object based functional programming paradigm find enough references depict supporting declarative language pro mainly used dealing databases allows developer connect database extract manipulate sql statements pro declarative statements mainly used execute sql statements following figure depicts eight declarative statements comparative studies programming languages within diverse criteria team term report figure taken compiling pro program two phases first phase compile pro second phase compile sql statements replaces suitable methods perform task javascript javafx built top java platform supports web applications javafx support desktops browsers mobile phones boxes javafx scrip language considered declarative language compiling javafx files result java byte code java byte code protable execute plateform suitable java virtual machine javafx software provide ability control playback videos ability comes api package mentioned api allows programmer built new features elaborate video player javafx technology allows programmer build customized graphical tool see capabilities javafx see javafx script allows programmer declare web api application using predicates describes desired gui components well relationship javafx script interprets components following example depects taken import import import import import stage title declaring easy scene scene width height content rectangle width height arcwidth archeight comparative studies programming languages within diverse criteria team term report fill stroke strokewidth circle centerx centery radius fill stroke strokewidth following picture shows result executing previous code batch scripting find resources talks batch script cpp found one cite talks script uses library header files uses dynamic programming cen compiled compiler scripting developer use dynamic typing declaring variable tag static typing using types dynamic typing may lead runtime errors following code script include cppscript var var args function called writeln hello world return comparative studies programming languages within diverse criteria team term report variable types script integers doubles strings chars booleans containers arrays maps lists etc objects methods functions iterators null script support operator casting comparison scope cloning control flow foreach statement user defined functions exception handling throw exception catch finally statements javascript javascript able detect add move change create delete read write remote files following jscript sample instantiate file system activex object var fso new activexobject invoke method var true something test close connection prototype design prototype means analyzing problem mapping existing solutions design producing simple design simulate solution improving design reach final result defiantly leads save money resources prototyping helps developer proof concept exploring design choose suitable language implement design provides many compilers gui designs prototypes activex allows developer use existing components internet activex controls helps use interent virtual environment atl active template library collection classes works mfc components com component object model technique used interprocess communications like remote method invocation java dcom distributed com glut opengl utility toolkit independent library used opengl programs gui graphical user interface information compilers extracted mfc provides help developers develop windows applications user interface applications considered fast technology create prototypes windows platform web browser capabilities reuse html windows components comparative studies programming languages within diverse criteria team term report javascript javascript support prototyping helps developer use concept simulate inheritance javascript programs objects prototypes inherent base object prototype fact makes adding new method instance simple task following example shows add method push object type array function var newarray inside function refers array var length length return newarray var examplearray new array inherits properties examplearray instance array object inherent new instance method push works depicted previous implementation javascript library called prototype library used dynamic web applications library aims reduce javascript coding prototype window class pwc javascript class provides developer ability easily generate api within html file following lines shows example api comparative studies programming languages within diverse criteria team term report generated windows inherent features windows gui components means maximize minimize move generated windows following figure shows prototype window class properties methods comparative studies programming languages within diverse criteria team term report javascript uses scripting api package includes following interfaces bindings compilable invocable scriptcontext scriptengine scriptenginefactory classes abstractscriptengine compiledscript scriptenginemanager simplebindings one exception class scriptexception detailed information classes interfaces found following cite aspectj default secure programming practices secure programming safe programming primarily based two important properties programming languages type safety memory safety fundamental characteristic guaranteed type safety strong typing type soundness ensuring one key issues protect system various security threats also requirement language garbage collection otherwise restrict allocation memory primarily language types interact according rules defined violated program semantics thereby ensuring type internal consistency specifically supports static typing allows perform type checking compile time moreover one distinguishable feature make certain part program dynamic typed via dynamic keyword clr garbage collector executes part program frees memory managed objects automatically longer referenced technique relieves programmers explicitly memory object listing test method executes array hold bytes allocated memory heap method exits local variable pops scope meaning nothing left reference array memory heap array becomes eligible reclaimed garbage collection free memory however possible force work garbage collector calling public void test byte new byte listing example memory allocation addition supports disposal pattern explicitly implementing idisposal interface dispose method released unmanaged resources like database connections handles files etc shown listing myclass implemented idisposable interfaces release unmanaged resources like file object class myclass idisposable filestream mystream object public myclass string filepath mystream new filestream filepath public void dispose comparative studies programming languages within diverse criteria team term report mystream finalize queue mystream listing example dispose method unmanaged resource hand aspectj also programming language also considered like base class java however researches revealed unlike java aspectj safe type system binding pointcut advice rise type errors runtime also aspectj typing rules severely restrict definition certain generic advice behavior aspectj concern memory monitoring management applied pointcut program better memory management approach used managed unmanaged resources files handles connections static typing programming language whereas aspectj really typesafe language sense concern applied better memory management aspectj could important application web application development web application kind application accessed network internet intranet usually hosted web server client side operated thin client web applications getting popular due ubiquity web browsers convenience using web browser client nowadays various platforms frameworks implemented web development primarily web application development required coding development two specific areas server sided scripting client side scripting client side development languages like javascript flash action script ajax etc popular server side development aspectj used framework platform language platform used web application development asp technology every aspx file class file contains class used asp programming implementation called code behind file class may contain initializes event handlers supporting methods codes etc developed web applications needed host microsoft iis server web development using performed visual studio ide provides extensive rich support development like web forms designer various built web server controls framework libraries etc development using flexible faster convenient programmers web application development aspectj separates concern core concern perform several activities research ibm ron bodkin implemented several monitoring checking error handling technique using aspectj implemented web application development monitor multiple applications monitor detect common application failure error handling repair database request monitoring authentication verification user session management comparative studies programming languages within diverse criteria team term report aspectj provides improved technique implement concern better way without hampering existing code comparatively better rapid convenient web development whereas aspectj better modular coding web services design composition web service software system designed support interoperable interaction network interface described able format specifically wsdl systems interact web service description using soap messages typically conveyed using http xml serialization conjunction standards basically web services based core set standards describe syntax semantics communication xml provides common syntax representing data soap simple object access protocol provides semantics data exchange wsdl web services description language provides mechanism describe capabilities web service unlike traditional systems web page web service provide users gui rather share business logic various processing interfaces across network framework provides extensive support interoperability web services using framework visual studio creating web service simple create web service project add public attribute webmethod expose creating web service simpler using framework visual studio ide compiling deployment web service easy faster simple web service example given receives message client returns web service using system using using using webservice namespace http webservicebinding conformsto public class service public service webmethod public string echo string message return message listing simple web service developed simple apache axis web service equivalent previous one created asp created using aspectj mentioned apache axis rapidly becoming popular web service implementations java developers also high scale success interoperability comparative studies programming languages within diverse criteria team term report web services frameworks apache axis running within tomcat server machine use following code compile deploy simple web service package public class mywebservice public string echo string message return message listing web service class web method package public aspect addmessageheaderaspect public pointcut captureecho string message execution public void string args message object around string message captureecho message return original message message listing web service aspect deployment http xmlns http service mywebservice java rpc parameter classname parameter allowedmethods listing automatic deployment comparative studies programming languages within diverse criteria team term report example taken russell miles basically aspectj perform various functionalities web service monitor multiple services hosted single server monitor web service call monitor service failure repairmen service pipeline creating web service aspectj comparatively tedious complicated required compile source file class file creating required automatic deployment service also additional work make service live tomcat server whereas case fast easy deploy service iis server aspectj separation concern make service modular aspectj separate concern core concern perform monitoring verification authentication providing notification without tangling scattering code abstraction programming oop programming paradigm uses data structures consisting data fields methods together interactions design applications computer programs oop programming techniques may include features data abstraction encapsulation modularity polymorphism inheritance development methodology basically four principles entities relationships must satisfy following four principles abstraction data functions principle means computation separated specifically entities consisted data functions functions permitted manipulate data entity directly whole entity manipulated unit one important principles paradigm information encapsulation principle means implementation information mechanism computational entity hidden computational entities entity required provide well defined interface entities interact purpose principle shield service using entity like class implementation details service provider entity inheritance principle means create child entity based parent entity child entity obtain properties parent entity new entity changed data functionality perform specific new task capability provides different ways utilizing reusing previously developed entities polymorphism methods polymorphism principle ability create methods similar entity specific functionality principle implemented conjunction inheritance principle comparative studies programming languages within diverse criteria team term report simple approach class myclass private int private access private int private access public int public access methods access member class access private members class public void setx int public int getx return public void sety int public int gety return class main class program static void main string args myclass new myclass access allowed setx getx methods accessed like wrong private comparative studies programming languages within diverse criteria team term report listing simple program listing see simple program satisfies basic principles properties paradigm like class method message passing abstraction message passing encapsulation etc hand aspectj aspect encapsulates implementations functionalities aspects similar classes many ways similarities discussed aspects include data members methods data members methods aspects role classes instance aspect manage state using data members whereas methods implement behavior supports crosscutting concern implementation utility methods aspects may also constructors included aspects access specifications aspect access specifier provides visibility like classes interfaces aspects public package access moreover nested aspects also similar nested classes public private protected access specifier aspects abstract like class aspect contains abstract pointcuts methods must declare abstract aspect abstract aspect mark pointcut method abstract refer constructs subaspect abstract aspect define every abstract pointcut method base aspect adds additional abstract pointcuts methods must also declare abstract following example shows abstract aspect contains abstract pointcut abstract method aspects extend classes abstract aspects well implement interfaces aspect inheritance properties one basic principles oop example following concrete aspect extends abstracttracing aspect able provide definitions abstract pointcut method matching requirements tracing banking system public aspect bankingtracing extends abstracttracing public pointcut traced execution banking public logger getlogger return banking comparative studies programming languages within diverse criteria team term report aspects embedded classes interfaces nested aspects aspects embedded classes interfaces aspect implementation closely tied enclosed class interface though aspectj two languages two different programming paradigms respectively key properties like classes aspects significant similarities aspect supports properties like abstraction encapsulation inheritance etc partially cases moreover aspectj developed extension java often called pure language undoubtedly supports four major principles programming basically aspect required separating concerns core concerns implemented using methodology eventually aspectj comes support programming paradigm oop aop reflection programming reflective programming functional extension objectoriented programming paradigm programming includes selfmodification however emphasis paradigm dynamic program modification determined executed runtime reflection programming language used observe dynamically modify change program execution runtime many classes support reflection part reflection api namespace thus program use reflection include namespace reflection namespace rich powerful support reflective programming extended programming paradigm static binding example represented dynamic use reflection without reflection string programming int length reflection object programming propertyinfo prop length getvalue get value propertyinfo int length int null listing reflection example languages aspectj offer powerful controlled mechanisms modify execution flow aspectj offers alternative way access static dynamic context associated join points reflection api example api access name currently advised method well argument objects method common use reflective information implement tracing similar aspects comparative studies programming languages within diverse criteria team term report aspectj supports reflection area powerful controllable reflection namespace based paradigm whereas aspectj reflection api paradigm programming aop programming paradigm secondary supporting functions isolated main program business logic aims increase modularity allowing separation concerns forming basis software development aop implementations crosscutting expressions encapsulate concern one place till release extensions implementation carried microsoft corporation programming paradigm currently several aop frameworks available space implemented third party approach positive negative attributes among renowned one provides comprehensive infrastructural support developing applications aop implementation aop implemented language need special compilation process weaving done runtime aop need control modify way assemblies loaded thus suitable use clr environment since support aop yet see example implementation authentication procedure response aspectj hand class program static void main string args sendmessage new sendmessage authenticator auth new authenticator try authenticate important message catch exception exception generated verification failure match message public class sendmessage public void sendnow string message message comparative studies programming languages within diverse criteria team term report public class authenticator public void authenticate string user string pass try input user password username user password pass catch exception user name password throw exception user throw new exception listing implementation authenticated message sending delivering message authentication method required invoke check whether user authenticated method requires authenticated call authenticate method leads code tangling similar code needed included classes require authentication public class main public static void main string args sendmessage new sendmessage important message public class sendmessage public void sendnow string message message public aspect securityaspect comparative studies programming languages within diverse criteria team term report private authenticator authenticator new authenticator pointcut secureaccess execution secureaccess checking authenticating user public class authenticator public void authenticate string user new string string pass new string bufferedreader new bufferedreader new inputstreamreader try username user password pass user pass catch ioexception pass throw new authenticationexception match public class authenticationexception extends runtimeexception public authenticationexception string message super message listing implementation authenticated message sending aspectj without changing single line code sendmessage class listing enhance functionality adding aspect system mentioned securityaspect authenticator class asks credentials username password authenticate method called aspectj example written help book aspectj action ramnivas laddad comparative studies programming languages within diverse criteria team term report two examples easily understandable concern like logging authentication make code tangling scattered whereas aop programming makes modular readable functional programming functional programming programming paradigm treats computation evaluation mathematical functions avoids state changes mutable data emphasizes application functions contrast imperative programming style emphasizes changes state programming functional style also accomplished languages specifically designed functional programming example higher lambda functions employed write programs functional style code example functional implementation taken release framework completely different coding style available functional programming could actually functional programming framework resulting code ugly hard understand glance hard maintain well lambda method syntax extension methods produce code written functional style listing convert method used trim empty space ends strings passing method defined using lambda syntax functional prgramming example lanbda expression stream stream str kvp count listing implementation functional style specifically aspectj functional programming implementation observed rather research works functional language implementation prototype implementations found aspectual caml aspectfun aspectml papers authors made approach build extension functional language create hybrid programming language reverse occasion basically programming paradigm developed top programming paradigm aop core concerns primarily developed approach since aspectj developed superset java holds functionalities properties java superficially java functional language however using interfaces inner classes fairly easy mimic important features functional programming using interface real functions possible write functions take functions parameters construct return new function addition open source implementations available functional programming java like functionalj functional java jfun etc next release java lambda expression introduced functional programming concepts like closure higher order function etc comparative studies programming languages within diverse criteria team term report comparison aspecj subset java provides functional programming ability built namespace library declarative programming declarative programming programming paradigm expresses logic computation without describing control flow many languages applying style attempt minimize eliminate side effects describing program accomplish rather describing accomplishing declarative programming often defined style programming imperative number common definitions exist attempt give term definition simply contrasting imperative programming example program describes computation performed compute programming language lacks side effects specifically referentially transparent language clear correspondence mathematical logic ability programming declarative style observed two new implementation linq regular expression linq language integrated query set language framework features writing structured queries local object collections remote data sources linq introduced framework enables query collection implementing ienumerable whether array list xml dom well remote data sources database tables additionally regular expressions notable implementation language able identify complicated character patterns linq implementaion var linqdata values rowline collection linq expression getting counts groups var found linqdata group select new names counts regular expression regex new regex exp find blood pressure reading used listing declarative programming specifically aspectj declarative programming implementation observed base class java declarative programming implemented annotations comparison aspecj subset java provides declarative programming ability built library linq regex comparative studies programming languages within diverse criteria team term report batch scripting operating system command line scripting capabilities core support systems administrators power users relatively unknown many users purpose make comparative study powerful utility command line aspectj comprehensive analysis performed identify capabilities various batch scripting techniques msdn rich powerful implementation namespaces used executing external commands example implementation external command process proc new process false false true listing example batch script extensive library support work microsoft office packages like reading writing creating files macros etc using method properties diagnostics namespaces monitor system performance send receive external help internal library support programming aop natural fit solving problems system monitoring aop lets define pointcuts match many join points want monitor performance write advice updates performance invoked automatically whenever process enter exit one join points research ibm named aop work performance monitoring aspectj basic performance monitoring system developed ron bodkin using aspectj system captures time counts different servlets processing incoming web requests approach applied implement batch script monitor system activities support java api external command execution unlike batch script aspectj written windows linux platforms moreover java api support command automation scheduling etc less like additionally aop implementation provide better modularity separating concern like security authentication statistics collection etc prototype design visual rapid convenient way create user interface visually using windows forms designer toolbox three basic steps creating user interfaces comparative studies programming languages within diverse criteria team term report adding controls design surface also dynamically required setting initial properties controls writing handlers specified events possible implement thread windows form program backgroundworker class intensive task needed done another thread avoided freezing stop responding implementation portability problem makes failed acquire important feature platform independence designing possible microsoft operating system since microsoft proprietary language help deployed platform although possible create writing code designers enable work much rapidly possible manual coding since aspectj extended java came user interface api provided java software development perspective distinguished properties supports availed features aspectj kind policy rules regulation update provided notification message without hampering core concerns system writing new aspect particular work memory management gui application calling garbage collector demand certain pointcuts example performance monitored aspect implementation synchronizations thread monitored respective pointcut whenever thread enters exits method java extensive api used implementing user interface ide convenient designing user interface extensive support tools controls provide flexibility faster development techniques programmers though designing aspectj java api little tedious less convenient provide additional flexibility concerns separating core concern making code modular haskell java default secure programming practices security primary concern language design implementation well define reliable program execution prevents attackers circumventing security policies exploiting weaknesses language models unified modeling languages provide way express higher level system abstraction language safety generalization common notions type safety memory safety advanced programming abstractions threads distributed message passing become common modern languages associated language features give programmers powerful flexible control various resources recent languages based security research development seeks add security behaviors language execution models features programming security policies language syntax comparative studies programming languages within diverse criteria team term report security java java language mostly used internet purpose hence security important networks provide avenue attack computer hooked controlling memory devices hardware strongly typed language helps reduce errors programs compile time enhances integrity security software automatic memory managementgarbage correctness garbage collector implementation essential reliability security java rmi collector widely used distributed garbage collector still difficult implement system rigorous safety real time requirements overheads incurred garbage collection much research scheduling garbage collector improving efficiency code transformation even though proven particularly effective far security haskell due concept memory side effects result pure expression used removed without affecting expressions thread safe strongly typed types may polymorphic may contain type variables universally quantified types haskell require explicit type declaration type interface system provides static type checking automatic memory managementgarbage haskell internal garbage collector web application development web application application uses web browser client web applications based architecture client enters information server stores retrieves information applications broken different layers known tires presentation layer side layer creates visual gateway consumer interact application range basic html dhtml complex com components java applets application tier range web scripting server side programming tcl corba perl allows user perform complex actions web interface database databases allow developers store retrieve add update categorical information systematic organized fashion comparative studies programming languages within diverse criteria team term report complex application solution may fall short tiered approach benefited break business logic resides application tier fine grained model web application java java builds high quality web application java apis create applications servlet component used controller java bean component model jsp used view template enterprise java beans used model located distributed environment java platform provides support security authentication authorization transaction database connection management jndi api provides flexibility java api xml processing jaxp part java platform supports processing xml documents using document object model dom jaxp enables applications parse transform xml documents independent particular implementation java persistence api java technology solution persistence persistence uses mapping approach bridge gap model relational database web application haskell haskell inbuilt components like servlets java create web application relies certain libraries cgi xhtml libraries monad transformers haskell used add application specific functionality database connectivity system hdbc library allows write code access data stored almost sql database web services design composition web services considered one kind service useful offers syntactical interoperability remote services platform independent way using web service technologies like soap simple object access protocol wsdl web service description language provide web service document uddi universal description discovery integration service registry web service interact web service matter platform web service developed run web service client many types another web service client written scripting language client java client etc web service composition java development java technology standards occurs java specification requests submitted java community process comparative studies programming languages within diverse criteria team term report java remote method invocation enables programmer create distributed java technology invoke objects java virtual machines two popular java web services architecture api xml based rpc standard way clients invoked simple java commands using service factory creation instances services access points implementing enterprise web services specifies web services java enterprise edition builds soap wedl cover use uses java naming directory interface jndi obtain service interface two instantiate local jndi context jndi lookup web service name context web service composition haskell web services performed integrating xml serializer producing consuming message representations based executor services haskellxml toolbox validating introduces general approach processing xml haskell toolbox uses generic data model representing xml documents including dtd subset document subset haskell data model makes possible use filter functions uniform design xml processing applications hayoo provides web service api used retrieve search result structured format abstraction abstraction mechanism practice reduces factors details one focus concepts time abstraction apply control data control abstraction abstraction actions data abstraction data structures control abstraction involves use subprograms related concepts control flows data abstraction allows handling data bits meaningful ways example basic motivation behind data types abstraction java java programs organized object normally consists visible non visible data fields methods java offers abstraction means abstract class type interface implementation details allowed information hiding naturally supported abstract classes used abstract data structure well functions behavior make cohesion data structure related behavior hierarchy classes built bases one choose reuse behavior code data structure bound behavior built associated data structure properties comparative studies programming languages within diverse criteria team term report program contains abstract method named check abstract class self written code import compare class declared abstract check int int abstract method abstract class class child extends compare class subclass check int int abstract method true output console abstract class abstract void void else false close check method child demoabstract program abstract function call void main string args int args child child creating object child class abstract method class public static args program executed command prompt java abs command line inputs output false abstraction haskell monadlab dsl embedded template become central programming abstraction haskell benefits including modularity maintainability comparative studies programming languages within diverse criteria team term report effective programming mathematical precision monads used support imperative features within purely functional language really provide modularity defining operation monadically hide underlying machinery way allows new features incorporated monad transparent manner abstraction performed help type classes provide clear separation data abstraction function abstraction type classes allow default implementation enables implementation generalization type classes instance declaration done outside type declaration site even outside whole module contains type type class substitute class type object one powerful abstraction mechanism available higher order function haskell function citizen freely passed functions retuned result function stored data structure program illustrates typeclass haskell decleration function isequal class class basiceq isequal bool defination function instance class instance basiceq bool isequal true true true isequal false false true input program ghci equal false false output program true comparative studies programming languages within diverse criteria team term report comparison java haskell abstract function declared abstract class defined subclass extends abstract class mandatory line code program compiled using command prompt class path path set per complete syntax command line arguments converted primitive data type prototype function takes integer nos haskell may look like objects object oriented programming haskell subtypes refer instance types letter instance type typeclass type implements functions defined typeclass typeclass defines one function function takes two corresponding instance returns boolean reflection computer science reflection process computer program observe modify structure behavior normally instructions executed data processed however languages programs also treat instructions data therefore make reflective modifications reflection commonly used virtual machine programming languages like scripting languages less commonly used manifestly typed statically typed programming languages reflection used observing modifying program execution runtime reflectionoriented program component monitor execution enclosure code modify according desired goal related enclosure reflection also used adapt given program different situations dynamically reflection java reflection commonly used programs require ability examine modify runtime behaviour applications running java virtual machine two main aspects introspection intercession abilities program observe modify respectively state behavior java reflection distinguishes static dynamic objects modified created read used mechanism provides partial evaluation ability examine environment package provide access representation java classes one tangible use reflection javabeans software components manipulated visually via builder tool tool uses reflection obtain properties java components classes dynamically loaded comparative studies programming languages within diverse criteria team term report reflection haskell reflection reflection ability program generate new code incorporate execution main use would allow abstraction types provided parametric polymorphism type classes structure types observed including names constructors fields types similar reflection api java attributes method signatures observed objects constructed class name approach handles regular data types nested data types data types constructor parameterised additional types handles single term traversal supports higher order generic programming reusable definitions traversal strategies overriding generic functions specified types generic functions directly defined haskell data types one way provide would extend language new constructs would require changes existing compilers interpreters would cause compatibility problems existing code way use separate tool provide linguistic reflection haskell uses separate tool derive written entirely haskell derive perform compile time reflection computing programming aop programming paradigm isolates secondary supporting functions main program business logic aims increase modularity allowing separation concerns forming basis software development aop includes programming techniques tools support modularization concerns level source code software development refers whole engineering discipline aspect orientation java aop concept bound certain programming language programming paradigm help shortcomings languages use single hierarchical decomposition may procedural object oriented functional java implementation aop called aspectj created xerox parc like compiler aspectj compiler includes weaving phase unambiguously resolves classes aspects aspectj implements additional phase first weaving aspects regular code calling standard java compiler comparative studies programming languages within diverse criteria team term report aspect oriented programming haskell ghc model aop style programming via simple transformation scheme aop programming idioms mapped type classes easily advise functions programs carry type annotations embedding programming style haskell provides structuring syntax directed compilers implemented attribute grammars convenient notation specifying functions deal production rules abstract syntax attribute grammar systems offer decomposition aspect syntactic level semantic level trex extension haskell provides rich set records trex stands records extensible records key component modularity approach define attribute grammar standard strategy writing attribute grammar consists three steps namely definition semantic domains semantic functions translators functional programming fundamental approach functional languages definition application functions allow functions treated values support higher order functions functions take functions parameters combine functions create new functions functional programming java java language sense executed without class superficially java functions however using interfaces inner classes fairly easy mimic important features functional programming using interfaces inner classes possible much pass functions parameters fairly easy mimic many features functional programming language program example reads transfers data one text file another text file uses input output buffer stream import import public class demofilejava comparative studies programming languages within diverse criteria team term report public static void main string args throws ioexception string str file new file data exist else fileinputstream fileinputstream byte byte char char string char file file fileoutputstream fileoutputstream comparative studies programming languages within diverse criteria team term report data written else file exist output file data copy data check file program compiled manner mentioned section functional programming haskell pure functional language free replace expression program resulting value without changing program meaning referential transparency makes possible reason programs correctness similar way would reason mathematical formulas every variable defined exactly modified later program read writes data file without side effects example illustrates combination monad system lazy evaluation function purity mentioned section abstraction like java implemented haskell way comparative studies programming languages within diverse criteria team term report file import import toupper main inh openfile readmode outh openfile writemode inpstr hgetcontents inh using hgetcontents methods required ever consume data input file hputstr outh map toupper inpstr hclose inh hclose outh hclose close instance file program map function defined haskell prelude order apply elements list comparison program code consists lines call function within function contains impure function functions enclosed within class contains side effects java sequential evaluation code small function contain returns result called haskell supports lazy evaluation required ever consume data input file using hgetcontents whenever haskell system determines entire string hgetcontents returned garbage collected declarative programming declarative programming way specifying program rather specifying needed prescribe computer steps take order rearrange program much freely maybe even execute tasks parallel declarative programming often defined program describes computation performed compute comparative studies programming languages within diverse criteria team term report programming language lacks side effects language clear correspondence mathematical logic declarative programming java java object oriented language integrated facilities supporting declarative programming providing library called jsetl programmers cant restrict constraints try provide comprehensive collection facilities support real declarative style programming declarative programming haskell haskell based models computation fundamentally different statemachine model underlying imperative programming languages haskell functional programming language based lambda calculus makes extensive use pattern matching encourages new extremely useful way programming haskell compiler automatically infers type expression thereby enabling catch type errors without programmer explicitly specify types datum batch scripting batch script allow several commands would entered manually command line interface executed automatically without wait user trigger stage sequence batch script java java class used create operating system process peocessbuilder instance manages collection process attributes java either load dll direct linking library contain external program implementation required native methods dll dynamically register native methods using jregister natives jni entry batch script haskell ghc package contains ghc api used module access external commands possible invoke external commands haskell raw system invoke specified program specified arguments return exit code program important process termination external program exit external command invoking haskell automatically indicates exit whenever program aborted exception haskell foreign function interface means haskell code use used code written language order call foreign function haskell import externally defined functions haskell either static linking dynamic linking comparative studies programming languages within diverse criteria team term report prototype design prototyping means exploring ideas invest often created early project planning specification phase developers write production code need exploration greatest time investment needed viable software web designers create users interact design real product prototypes easy inexpensive create conventional purpose prototype allow users software evaluate developers proposals design eventual product actually trying rather interpret evaluate design based descriptions prototyping also used end users describe prove requirements developers considered commonly used prototypes windows based gui gui development java user interface libraries used java heavyweight native abstract window toolkit awt provides gui components means laying components means handling events components lightweight swing libraries built awt provide implementations awt widgetry apis audio capture processing playback gui development haskell several toolkits available haskell popular commonly gui library haskell based gtk extensive multiplatform toolkit creating gui interfaces fundamental thing widget widget represents part gui may contain widgets examples widgets include window dialog box button text within button wxhaskell portable native gui library haskell wxhaskell built top wxwidgets comprehensive library portable across major gui platforms including gtk windows macos mature library development since supports wide range widgets native wxhaskell consists two libraries wxcore wxcore library provides core interface wxwidgets functionality using library like programming wxwidgets provides raw functionality wxwidgets php scala default secure programming practices evert program insecure security important factor way programming languages designed maintained every programmer know avoid critical security mistakes comparative studies programming languages within diverse criteria team term report performing reviews testing security bugs measurement characteristic security must balanced expense usability must part php one striking things php programming language beginner programmers achieve simple goals rather quickly problem hand many programmers aware happening backend known security sacrificed sake convenience php flexible file handling functions inclue require fopen functions accept local paths remote files lot vulnerabilities due incorrect handling path names another problem php writes variables global scope indeed advantages however get lost big scripts egpcs environment get post cookie server variables put global scope one concept must always remember user input unreliable trust quickly examples input validation partially lost transmission server client corrupted process modified user unexpected manner intentional attempt gain unauthorized access crash application extremely important validate user input processing programming plain php rather boring without sql connection one problem sql queries unchecked variables dangerous course javascript validation validations entirely useless since easily bypassed register globals directive automatically make variables environment get post cookies server data criticized lot however security vulnerability risk bad practice result disabled default since version used instead php manual contains great section especially security precautions coding php scripts manual notes possible security risks exist prevent minimize side effects validating input sql queries ultimately creates vulnerabilities sql injection validation user input cookies creates xss cross site scripting vulnerabilities biggest problems php also scattering sql queries php codes create mess scala scala thought less secure java superficial ways mostly related visibility members outside classes terms helping write code free security holes better follow best practices immutability concurrency etc jsp kind domain specific language bad mixture html xml java scala programming language jsp template language compiles java servlets allows arbitrary java code snippets scala programming language java usually code size scala basically java respect common programming practices scala programming language used many companies develop commercial software production system edft twitter xebia xerox foursquare sony siemens gridgain appjet reaktor many others comparative studies programming languages within diverse criteria team term report web application development php php great web applications php scripts executed server php supports many databases mysql informix oracle sybase solid postgresql generic odbc etc one main reasons php popular open source software free download use php released php license php combined mysql develop windows serve unix platform php runs different platforms windows linux unix php compatible almost web servers used today apache iis php easy learn runs efficiently server side explains many new programmers tend program php stick knowledge language however rise web programming web uses mostly ajax javascript xml css also rise html make php difficult position server side language ajax later html key web applications scala lift one albeit popular scala web framework play another one people seem like lift free web application framework lift aims deliver similar benefits ruby rails except lift applications written scala instead ruby use scala means existing java library web container used running lift applications lift pretty much java development lift programmers use standard java environments like eclipse idea lift written scala programming language modern language java virtual machine java higher version needed developing running lift projects well suitable versions scala libraries compiler needed lift depends servlet api hence need suitable servlet container run web application jetty tomcat scala lift code brief expressive ruby code lift offers developers amazing productivity gains versus traditional java web frameworks rails hand lift code scales much better rails code lift code compiler becomes friend lift web framework awesome ajax comet support powerful concise rails scalable secure production web sap web services design composition web services typically application programming interfaces api web apis accessed via hypertext transfer protocol http executed remote system hosting requested services web services classified two categories big web services restful web services web services allow exchange information http using xml example want know weather another city write short script gather data format easily manipulate developer perspective calling local function returns value comparative studies programming languages within diverse criteria team term report php web service consists server serve requests web service client invoke methods web service php class library provides soap extension develop soap servers clients extension create servers clients important advantage web services ubiquity across platforms languages php script running linux talk iis server windows using asp without communication problems server switches solaris apache jsp everything transitions without glitch soap popular web services format standard passing messages across network calling functions remote computers php come bundled soap extension begin need download install files let easily integrate soap applications simple use soap wsdl php clients allow gather information across net use scripts major company provide soap front end data google lets search listings times day additionally xmethods large directory soap servers experiment use appendix create basic web service provides xml json response using php mysql scala building web services lift somehow easy pattern matching functions scala xml support lift support rest web services appendix building web services lift abstraction php new features appear new version released existing features improved php object support one feature improved version object oriented support first appeared php php made additional improvements way constructors handled php object support growing many reasons developers might take object oriented approach diminishing started really interesting php appendix example creating simple abstract class called oophpabstractclass oophpclasstoextendanabstract extends scala java abstract methods pass method parameter abstract fields pass value parameter similarly abstract type members specify type parameter comparative studies programming languages within diverse criteria team term report scala team decided construction principles three sorts members abstract fields well value parameters pass methods parameters abstract specify types parameters abstract model one terms express every sort parameterization form objectoriented abstraction sense scala orthogonal complete language reflection php comes complete reflection api adds ability classes interfaces functions methods extensions additionally reflection api offers ways retrieve doc comments functions classes methods parts internal api missing necessary code work reflection extension example internal php class might missing reflection data properties considered bugs discovered fixed external libraries needed build extension installation needed use functions part php core extension configuration directives defined api takes language introspective abilities far mature stage includes convenient methods permit developers dissect classes interfaces bare bones useful appendix reflection example shell terminal scala scala support reflection java scala richer types fully reflected bytecode scala reflection library works scala reflection built top java reflection reassemble view program damaged view returned java reflection implements abstract api shares scala compiler recreate original scala view program class files api backend scala reflection library frontend ties backend java reflection userfriendly system scala reflection scala reflection library development time usable version available languages usually implement api supporting operations reflection however reflection apis generally follow three design principle reflection encapsulation stratification ontological correspondence scala specific api supporting however since compiled java compatible complete java library consequently one use java reflection api order access information scala program anyhow approach presents important limitations raises usability problems call object methods reflection comparative studies programming languages within diverse criteria team term report val val println classof null get reference singleton object val val console module null aspect oriented programming aop concept created originally java developers developed compiler implements aop white box approach aop methodology meant implement new aspects software component using external components without altering code implements core functionality php approaches facilitate programming php phpaspect uses compiler written php performs static weaving using source code transformations downside approach advantages stem php interpreted nature lost php uses preprocessor php written java responsible weaving due java implementation approach integrate seamlessly php platform aspectphp reimplementation php available patch extension php aop library php requires manual changes aop library php implemented implement aspect oriented programming executing code classes enable orthogonal aspects aop library php package implements framework provides php solution rely stage therefore used right away without eventual complication compiler based aop implementations bunnyaspects another implementation aop inside pure need extensions uses existing qualities abilities language carries set problems essence bunnyaspects wrap bunnyaspect object around target object bunnyaspect object keeps track woven magic method intercedes method calls calls advice needed comparative studies programming languages within diverse criteria team term report scala usually less need aspects scala since language powerful expressive like traits composing behavior implementation aop scala scala supports attributes could say scala support aop via scala also supports mixins enable separation code crosscuts class hierarchies related introduce member aspectj advice scala knows closures compiles closures inner classes bytecode level creates multiple bytecode classes one class one would mentally transform scala bytecode java order know write pointcuts advices scala code functional programming essential approach functional languages definition application functions allow functions treated values support higher order functions functions take functions parameters combine functions create new functions php rise javascript languages like python ruby functional programming becoming mainstream attempt define lots useful functions php fix things inconsistent others already supports things php already exist adds foldr compose zip andf orf every curry flip new short hand way define functions strings virtually documentation little way examples tests scala scala programs execute java virtual machine jvm interoperate java programs application programmer interfaces apis multiparadigm programming language natively supports imperative functional styles programming scala functions values functions objects take functions arguments return result anonymous may curried take arguments one time allowing partial application often passed closure references free variables manipulate provide ability build powerful libraries functions hybrid language scala require functions pure require variables immutable however encourage write code way whenever possible comparative studies programming languages within diverse criteria team term report robert fischer mentions functional programming language statically typed object oriented language rich hickey mentions scala really functional core argument made people scala functional language goes like mutable variables citizens language uncontrolled ties first point mutable collections imperative libraries exist equal footing structures class inheritance overloading etc verbosity declarative programming php modern programming languages php included imperative programmer describes computer perform particular task contrast declarative programming languages philosophy goes computer told programmer wants left computer declarative programming languages often considered somewhat exotic declarative language seen nearly every day application developers sql sql programmer formulates query left database engine query analyzer figure combination disk reads index lookups functions necessary satisfy query php declarative programming language however prevent programmers adopting declarative style programming consider code fragment piece code states users users particular permission access page far readable equivalent imperative style would involve explicit statement test user logged correct position consider code used read query parameters get parameters display time fields fields order orders orders currency currencies writing code way see glance type data expected parameter default value use parameter omitted style helps make code comparative studies programming languages within diverse criteria team term report documenting declarative style common many modern web development frameworks notably ruby rails scala scala list comprehensions provide declarative syntax abstracting data hides less declarative function call closure creation details using operator overloading scala possible also write decent prolog code declarative programming style scala makes software product development deployment enjoyable important note declarative reading math definition definition factorial integer times factorial def fact int int fact definition isprime integer given range integers thru modulo zero def isprime int forall batch scripting php cron module allows run commands predetermined times intervals cron normally available unix linux distributions daemon allows schedule program script specific time execution using cron automate many tasks example update content website generate quota reports remove expired articles send given date lot important aspect cron sends error output specified address debug problems occur cron driven crontab configuration file specifies shell commands run periodically given schedule crontab files stored lists jobs instructions cron daemon kept users individual crontab files often system wide crontab file usually subdirectory system administrators edit line crontab file represents job composed cron expression followed shell command execute cron expression string comprising fields separated white space represents set times normally schedule execute routine following cron configuration run php script day username first five fields define times script executed comes name user used run script rest line command line execute need comparative studies programming languages within diverse criteria team term report know php scripts time fields minute hour day month month day week command username runs script mark every hour command username runs script saturdays day week specified host cpanel may cron job interface take commands option allow configure visually times execute scala scala program may also run shell script respectively batch command bash shell script containing following scala code shell preamble exec scala object helloworld def main args array string println hello world args run directly command shell important note file execute access search path scala command specified path environment variable use scala replacement console scripting languages batch windows scala package contains like ruby python console test code snippets unlike java also real scripting scala advantages larger programs later even java projects access complete jdk put classpath use script file operations high object oriented level much easier writing bashscripts comparative studies programming languages within diverse criteria team term report programming get small readable scripts script gets advanced use create gui script one downside using scala scripting language every time start lasts seconds start majority taken time needed loading compiler compiling start jvm however using flag scala save compiled version script file load version instead however still fast perl python prototype design php set language bindings php allow gui applications written php provides interface classes functions originally inspired pygtk show could done php web anymore need gui application use favorite language fully utilizes php powerful object model support brings improved portability gtk well new set widgets project also new extensions gtksourceview provides rich source editor widget alongside old favorites documentation filling rapidly several articles tutorials written topic around half classes fully documented scott mattocks active member documentation group also written book pro php gtk apress subject programming platform independent scala gui applications developed based scala library provides access java swing framework gui classes wrapping library approach scala library resembles underlying swing classes hides much complexity scala wrapper based original java swing framework scala wrappers resemble underlying swing classes try simplify concepts possible make uniform simplification makes extensive use properties scala language everything object philosophy makes possible inherit main method gui application scala functions pattern matching make possible formulate event handling reactions component simple swing application scala import object firstswingapp extends simpleguiapplication def top new mainframe title first swing app contents new button text click comparative studies programming languages within diverse criteria team term report scheme bpel default secure programming practices author describes language safety general concept includes type safety memorysafety language implementations enforce program intended semantics achieved static type checking dynamic check type checking checking trap nonsensical operations scheme example safe language unchecked unsafe operations required certain scenarios explicitly defined language abstractions dynamic type checking supported type tags differentiate structures allocated heap types provide context operations underlying system selects correct operation limit semantically valid operations catching inadvertent logic errors observed nominative type systems names distinct declarations determine variable type better safety structural type system allowing coincidental structural matches among types example memory safety feature automatic garbage collection prevents occurrence dangling pointers memory leaks double deletes racket scheme dialect implements two kinds garbage collection generational garbage collector standard used less expensive allocation objects cgc conservative garbage collector allows interaction programs expensive slower example memory safety capability access control another language safety concern portability language completely defined manual abstractions language provides safe knowledge details language implemented manages memory order object necessary order write safe program scheme language manual example specifies exact behavior programs written language although scheme dialects maintain language manuals well documented comply standard scheme core run unix max windows platform however implement libraries addition standard libraries language security important observed many security violations due vulnerabilities language design implementation safe language features lists modifications access checks security distributing unforgeable references sensitive objects specification temporal properties allow communication protocols resource usage specification enforcement control execution environment controlled access local resources cpu files etc prevent attacks sandboxing processes among others sandboxing provided racket weaknesses mentioned scheme security project related eval language construct security guards racket provide explicit checks plt mzscheme also offers several fundamental security mechanisms custodians manage resource allocation security guards inspectors controlled access opaque structures namespaces control access scheme bindings thread groups control allocation cpu found racket well safe language technologies set ground building secure systems example web server built plt scheme currently known racket allow buffer overflow attacks security policies enforcement supported secure language technology type systems type checking play role abstraction well author observes interfaces modules seen types providing summary facilities exposed client modules participating composition contract implementers comparative studies programming languages within diverse criteria team term report users author discusses type safety xml documents enforced schemas define structure xml documents validated schema definition oasis standard web services business process execution language version requires conformant implementations enforce conservative static analysis bpel processes analysis described standard detect undefined semantics invalid semantics within process definition detected schema validation xsd static analysis includes checks performed wsdl interface also partners invoked process among many others bpel standard provides schemas language constructs addition since every bpel process invoked must described wsdl interface makes available clients bpel processes typed according observation given moreover bpel defines mechanisms security protection checking modified forged messages transit residing destinations detecting invalid expired messages support timestamps signed messages protection attack searching current bpel implementations found implemented using java language java described safe language bpel implementations possibly inherit properties however writing investigate mechanisms safety requirements implemented web application development web applications programs accessed browser clients human access applications functionality stored server side via browser businesses building web applications provide access software hosting maintaining allows distribute software products without need install applications client side business model leasing software solutions clients known advantage applications maintained centrally application service providers asp software evolves clients directly affected disadvantage inflexibility software adjusted satisfy clients needs better software product accepted addition access application done browsing manually initiating different functionality human user services provided asp difficulties integrate applications make interoperable incapacity customize software better fit customers needs led development web services built model allow development complex applications composing existing existing technology allows build web applications presentation layer browser java javascript dhtml flash silverlight application layer asp cgi coldfusion php perl python ruby rails http communication protocol used web applications scheme bpel scheme implementations allow programmer build web applications example racket implementation scheme example web application implemented using scheme bpel found appendix application accepts number user returns many strings input output scheme server come client browser bpel application requires schema describe input output represented xml document web page created example however would require use jsp technology servlet example dispatch request browser bpel process comparative studies programming languages within diverse criteria team term report similar web client functionality scheme achieved using technologies jsp servlets javascript possibilities addition scheme application modified add number obtained user return double changes required one line code computes response lack space functionality implemented bpel second functionality would simplified bpel application scope needed compared bpel scheme web server application programmed lower level explicitly manipulates tcp connections threads creation http pages build response handling security issues examples lot research effort directed building secure web servers scheme collection papers reporting accomplishments found bpel operates higher level abstraction bpel processes require web server deployed run although bpel verbose language built top xml technology existing implementations provide graphical ide capable source code models ide provide wizards generate wsdl interfaces schemas dtd xsd schemas created using graphical environment well security transport protocol handling handled implementations defined standard build web application scheme requires expertise experience language effort compared building bpel process web services design composition web services applications developed primarily enable automated interaction programs uses web services technology permit development complex business processes transactions services allows users customize business solutions better fit needs something deficient web applications programming model web services applications built components hiding implementation details behind standardized web service interfaces wsdl allow access functionality standardized communication protocols soap http exchange data via standardized wire protocols xml web services atomic composite atomic web services provide simple cohesive business prospective functionality composed complex business processes composite web services web service compositions expose functionality using web service interfaces customized solutions built either composing web services statically static binding protocol design services dynamically dynamic binding protocol late binding run time depending business requirements web services stateless keep state distinct operation invocations inherent underlying transportation protocol http also stateless order process request state needs retrieved persistent data repository web services also stateful transactions session conversation spans multiple requests composite web service preserve transient data conversation interaction pattern synchronous asynchronous one way messaging style services message contains individual parameters operation services messaging style document style whole document sent purchase order receiving service web services roles interaction web services web service providers web service requestors also fulfill roles given conversation web service providers describe functionality properties clients invoke services via wsdl interfaces advertise capabilities publishing wsdl interface service registry clients search services satisfy business needs web service client finds best web service provider registry bind using information web service interface comparative studies programming languages within diverse criteria team term report web services scheme bpel oasis standard built fulfill role composite web service capable holding state detailed tutorial build bpel application provided details components required build process including creating xml schema wsdl bpel process composite application service assemblies test composite application although scheme language implementations supporting development web servers applications literature found discusses scheme language support web service development possible reason web services platform language neutral existing languages implement web service functionality wrapping wsdl interface clients invoke functionality without aware implemented independence achieved using web service technology stack described detail abstraction evolution programming models languages supporting based classes objects said object oriented three fundamental concepts paradigm encapsulation inheritance dynamic method binding evolutionary paradigm building upon earlier ones earliest languages introduced features abstracted underlying platform made programs easier understand concepts important language design names used replace concepts machine instructions addresses scopes supported scoping rules determine program text name visible used bindings association made different language features binding time associations made compile load runtime referencing environment set active bindings provides context program given point aliases referring object given scope one name overloading referring one object given scope one name depending context reference made polymorphism object one type depending execution history context classes data abstractions encapsulate representation data provide operations manipulation evolution data abstractions began global variables visible parts program therefore modified executable statement program bugs difficult find variables exist whole duration program increasing program footprint local variables introduced reduce visibility segment program mechanism provides fault containment limiting program text fault might occur variables allocated memory time segment executing deallocated segment completes static variables objects period allocated computer memory visibility separated extent static variables whole program execution visibility controlled scoping rules next stage evolution data abstractions introduction modules encapsulate set subroutines share set static variables modules one two kinds module module kind exports abstract type variables record set subroutines type instantiated module users passed reference module subroutines kind allows creation instances whole module module abstract type including variables subroutines comparative studies programming languages within diverse criteria team term report paradigm extends notion adding inheritance form reusability mechanism distinct composition dynamic method binding allows functionality actual object memory executed context base type functionality expected encapsulation mechanism refined providing different levels visibility private protected public allow finer control names exported local class programming languages variations languages allow classes nested nesting useful creating object closers becomes mechanism capable capturing explicit representation referencing environment context later point program context bound subroutine allows separation subroutine referencing environment provides greater flexibility subroutine execute context created earlier classes hold variables reference variables point objects classes reuse composition programming languages vary provision reference model value model variables value model dictates implicit creation automatic variables program stack reference model objects dynamically allocated heap explicitly initiating construction destruction either explicit via language construct via language mechanism garbage collection languages type polymorphism support subtype polymorphism achieved using inheritance underlying mechanism implicit parametric polymorphism realized allowing variables bind type object value runtime explicit parametric polymorphism generics language feature allows types variables instantiation type variables binding type variable concrete type usually done compile time scheme language scheme language consists small set base constructs variables constants conditionals assignments procedures functions applications constructs implementing variable definitions procedure parameter specifications function macro definitions specific scheme language dialects plt scheme currently racket drscheme extend base language constructs modules generative structure definitions expressive macro system class systems classes plt scheme first class values lexical scoping rules single namespace mixins explicit inheritance implemented macros claims made objects instances created class system space time efficient compared java object smalltalk efficiency compared smalltalk java method invocations abstractions discussed context racket dialect scheme class expression racket value syntax class class root class object method constructor instead initialization expressions addition class hold declaration methods fields classes instantiated new form initializing arguments need given values instantiation time referenced directly methods used initialize fields class fields available methods class instances named anonymous racket provides inheritance mechanism expression used initialize super class initializing arguments field declaration supernew placed order class however relative order determines order evaluated class instantiated dependencies need ordered methods declaration order important fully defined class instantiated comparative studies programming languages within diverse criteria team term report interfaces racket used able check whether object class implements specific set methods declare class implements interface class form used class implement methods declared interface extends runtime error occur evaluation example class declaration shown figure object instance class created initialized field updated runtime value displayed figure class fish instance charlie racket bpel language authors comparing oop paradigm services oriented computing paradigm similarity differences concepts abstractions example think objects providing functionality services hiding implementation details web services describe objects services terms interfaces provide clients objects services oop method invocation contrasted soc shared execution context oop compared multiple contexts soc web services run interoperate heterogeneous environment middleware operating system platform language transparent soc model another difference interaction components oop distributed counterparts soc oop using synchronous method invocation communication mechanism contrast distributed nature soc necessitates asynchronous message passing achieve flexibility composition central soc oop design activity architecture system static many patterns created assist programmer build better solutions business problem contrast soc composition services dynamic web services built services dynamically discovered composed since architecture dynamic constituents web comparative studies programming languages within diverse criteria team term report services selected replaced based properties qos architectural viewpoint major shift designing static architecture oop providing infrastructure enable support dynamic selection composition distributed components soc bpel one languages fulfill objective authors propose thinking inheritance special form composition interface behavior composed object preserved opposed typical composition interface enclosed object wrapped filtered enclosing object comparisons drawn composition web services soc polymorphism oop ability operation take different operands objects different classes provide different implementation behavior depending operands types able build system uses polymorphism minimum requirement language formal type system according typing inheritance formalized notion polymorphism oop abstractions encapsulation information hiding hiding implementation details stable interface protect system changes relevant soc well reflection reflection mechanism programs perform computations means inspecting internal structure types introspection altering structure behavior program intercession mechanism allows programs query program symbol table run time inspect execution stack reflection useful provide performance statistics debugging tools among others languages support reflective programming need following characteristics incorporate facilities keep information program structure api observing modifying program execution runtime criteria assessing language support reflective programming include source code discovery modification runtime conversion string language construct evaluation string expression runtime interpreter creation give new meaning programming construct reflection scheme bpel language characteristics scheme language homogeneity program data scheme programs data manipulated using language mechanisms addition scheme operational semantics defined scheme interpreter written scheme language scheme capable selfdefining scheme dynamic language expand reflection mechanisms incorporated existing implementations scheme examples provided racket implementation scheme figure demonstrates reflection scheme code snipped adapted eval function takes quoted expression argument evaluates scheme lexical scoping rules capability eval dynamically evaluate expression eval see local bindings context evaluation attempted make set bindings visible available dynamic evaluation namespace mechanism used namespaces encapsulated within module access current module reflective hook needed namespace defined within current module using namespace example figure shows comparative studies programming languages within diverse criteria team term report figure scheme reflection example current programming model bpel based assumption information service composition remain static throughout lifetime consequence language support reflective features authors propose reflective framework extension bpel allow dynamic changes composition based changes environment reflected data processed authors goal reflective framework allow creation adaptive service composition reliable control flow data flow correctness proposed framework uses computational reflection mechanism structured two levels computations keeps information structures used framework data table containing participants declarations data declarations pfg parallel flow graph representing control flow explicit synchronization parallel programs operations framework provides adding deleting nodes synchronization edges two parallel branches data variables participants replacing nodes type reordering nodes experience creating systems different application domains showed using procedural functional logic programming techniques insufficient clearly capture implement certain design decisions problem domain operationalization comparative studies programming languages within diverse criteria team term report requirements forces functionality scattered across different components results code difficult develop maintain programming allows functionality identified isolated composed reused similarities programming computational reflection metaobject protocols specifically possible exploit reflective facilities language develop aop prototype systems reflection used tool model transformation another approach achieve goal aop different source models mde terminology transformed target model using model transformation definitions written language discussion seen oriented languages tools achieve goal aop report criteria language dedicated support specifying aspect pointcuts advice aspect compiler aspects base language provision programming environment implies conceptual model declaring aspect specifies pointcuts selection join points advice adding functionality base model expressible language aspect orientation scheme bpel language authors propose semantics advice weaving extending hindleymilner type inference system able applicable advices lambda expressions plt scheme extended features pointcuts advice functions using powerful macro system language extend also provides extension scheme language resolve issues dynamic scope generally needed aop languages extension bpel allows specification crosscutting concerns logging persistence auditing security support dynamic adaptation composition runtime authors use extension bpel build transparent security mechanism around processes scheme bpel aspect orientation implemented extension base language functional programming theoretical roots functional programming traced back lambda calculus computation done substitution arguments functions author states program regardless computation paradigm used seen proof proposition given appropriate inputs exist outputs related inputs particular desired difference stems way paradigms express computation relationship among abstractions example functional programming computation expressed mathematical function inputs without side effects typical features functional languages functions capable accepting functions parameters function values allowing functions threaded value program assign variables store structures passed returned function implicit parametric polymorphism types variable universally quantified types values hold determine type structured types lists constructors build structured types also called aggregates garbage collection automatically reclaim memory dynamically allocated data case stack allocation unsafe garbage collection essential functional languages since variables generally allocated space heap unlimited extend necessary case referencing environment needs saved closure retrieved later used creation passing functions parameters functions comparative studies programming languages within diverse criteria team term report implementation needs guarantee memory allocated variables accessed later program released ensure program run memory functional programming scheme bpel language scheme possesses characteristics functional language addition characteristics typical languages include homogeneity programs data represented lists run time capability languages lisp family express operational semantics terms interpreter expressed given language consequently given language capable program snippet demonstrates functional characteristics scheme provides scheme implementation computing factorial number using tail recursion function evaluated racket environment using drracket figure figure factorial function implemented scheme using tail recursion output test run approach assess whether bpel considered functional language compare characteristics functional languages bpel main characteristics functions function available language feature addition bpel executed keeps state control structures language imperative style examples loop sequential flow support parallel well assignments however comparison performed authors bpel another functional language found existing literature structure used implement factorial function bpel initially input value assigned output greater one scope entered value input decremented one multiplied output another attempt made enter scope outline control structure one used building web application example details provided appendix difference comes assignments computation predicate termination condition size source code sample input output provided appendix algorithms scheme expressed concisely without heavy supporting infrastructure required bpel scheme implementation factorial function purely functional comparative studies programming languages within diverse criteria team term report side effects updates variables paper possibility bpel recursive investigated implementation factorial algorithm using recursion requires sequence updates input output variables bpel language meant stateful web services composes stateless although implementation side effects updating underlying data repository declarative programming literature declarative programming defined informally description computed computed seminal article robert kowalski separates definition algorithm two concerns logic component defines knowledge used solving problem control component defines strategies using knowledge logic part algorithm determines meaning control part efficiency declarative programming defining logic component underlying execution engine implement control component others view declarative programs theories logic computations deduction theory languages based logic pure prolog logic haskell scheme qualify according definition view broad includes logic relational functional programming among applications formal methods theorem proving algebraic specification program synthesis according declarative programming subdivided weak programs theories complemented possibly control information programmer improve efficiency strong programs theories system supplies control michael hanus similar view declarative programming style programming describes properties problem domain solution rather steps computation needed obtain solution describes similarity classes depending underlying formalism programming languages support declarative paradigm description languages combine logic functional programming provided declarative programming scheme bpel language according description constitutes declarative language scheme qualifies since functional language included definition declarative languages bpel allows specification operational behavior descriptions uses programming language constructs describe process behavior different formalisms used specify system behavioral model based automata theory specification expected properties based logic declarative nature dynamic interface view business process described via protocol set legal conversations process outside world description process interface behavior purpose providing declarative specification interface protocol verify conformance concrete process implementation specification bpel abstract process specifies business protocol however since bpel imperative process language abstract specification authors extend bpel declarative behavioral specification language based form regular expressions batch scripting generally programs seen functions produce result manipulating input programs built fulfill specific purpose goal practical systems need configure coordinate discrete functionality implemented programs prescribed way fulfill higher business goal achieved using generalpurpose programming language since language features necessary control comparative studies programming languages within diverse criteria team term report flow sequencing selection iteration recursion variables data types subroutines control data abstractions automate coordination independent programs however purpose provide features security mechanisms static type checking built language support programmers creating efficient maintainable portable programs main requirements scripting languages contrast flexibility local customization rapid development dynamic type checking extensive use structured types tables lists files operations pattern matching string manipulation easy access system facilities economy expression requirements change emphasis feature set provided scripting languages batch scripting opposed interactive commands processing process specifying commands command interpreter file batch file consequently executed interpreted usually line line main purpose automate process needs performed regular bases batch scripting scheme bpel language searching existing literature provided many examples scheme language used scripting language online tutorials showing write scripts within specific implementation language necessarily compatible implementations scheme examples parsing using racket implementation scheme provided scheme program converted executable script unix mac replacing language declaration lang racket top definition area racket file permissions need changed using command line make new program executable similarly batch files created windows example figure figure show batch file output produced run command prompt command execute script written semicolon operating system automatically attaches path script argument command racket implementation provides command parse arguments supplied script script takes string argument flag flag long message produced otherwise input argument echoed back user figure batch file execute racket code windows batch files comparative studies programming languages within diverse criteria team term report figure output running file command prompt authors describe implementation bpel application executes script commands current run time environment using embedded java commands process cmdstring describe possible way invoke bpel process sell script able monitored controlled environment since bpel engine requires java server needs remotely invoked virtual machine launched sell script different using current technology proposed solution client intermediary shell script bpel process handle remote invocation possible exceptions thrown server authors illustrate requirements job control language describe complex grid applications dependencies error handling failure recovery monitoring resource release among others compare existing languages support job control grid applications orchestration bpel one languages satisfy requirements set forward example explain job orchestration implemented using bpel specific grid applications task handlers job submission file transfer achieved using partner setting threshold failed tasks tolerance task execution easily satisfied although powerful expressing error handling communication requirements would complex using bpel demonstrated possible execute external bpel using java api requires extensive knowledge bpel java api implementing capability addition bpel program footprint larger scheme batch script footprint reason scheme language explicit support batch scripting case bpel prototype design user interface development tools need support use management components organizing arranging layout capabilities functionality testing prototype interfaces minimum programming language used specify programming scripting user interactions affect usability tools different phases software development require different tool support user interface prototyping early stages requirements elicitation end users need closely involved design tools support target addition developers specialists graphical design necessarily expert programmers implementation phase tools need support gui components management usage allow creation prototypes set features capabilities prototyping tools need provide therefore different depend stage software development application requirements comparative studies programming languages within diverse criteria team term report prototype design scheme bpel language author introduces mysterx scheme toolkit capable building interactive applications achieved using reflective mechanism integrate host com components capable creating visual interfaces microsoft visual basic create dynamic html using javascript vbscript window components properties set using scheme language scheme used create event handlers respond user actions mysterx combines different technologies languages enable creation gui applications given description implementation scheme language allows programmers build graphical interfaces provides support core windowing classes geometry management defining containers called containees defining new types containers mouse keyboard events windowing class reference window area canvas button frames check box window widgets many capabilities similarly racket implementation scheme provides implementation gui design capabilities addition capabilities build dynamic web pages demonstrated bpel designed support composition web services using communication however support human interactions support human interactions proposed extension bpel language enables definition client applications support human interactions interaction bpel also achieved via cases however meant used production bpel deployed server invoked remotely wsdl interface external client exist another bpel process browser web service comparative studies programming languages within diverse criteria team term report consolidated analysis synthesis results criteria default secure programming practices web applications development web services design composition abstraction reflection close operating system especially relates pointers memory access fact makes unsafe creating web applications using toolkit web services use xml remote procedure protocol xmlrpc simple object access protocol soap transport layer send request receive response supports object oriented based programming provides multiple inheritance exception handling well interface definition static dynamic binding support runtime reflection support reflection javascript javascript dynamically typed considered unsafe javascript imperative scripting language mainly used client side internet applications javascript call web services client using java api web services framework helps web service javascript support inheritance virtual functions javascript build top java extendscript construct facilitates reflection behavior php dynamically weakly typed language garbage collection many functions defaulted extremely popular efficient fast free opensource many books php manual provides soap extension develop soap servers clients objectoriented pdo uniform access database criticized supports complete reflection api reverse engineer classes interfaces comparative studies programming languages within diverse criteria team term report true popular easy learn supports many web servers dbms crossplatform portable extension create xmlrpc servers clients simple use functions methods extensions scala static strongly typed uses type inference deduce type garbage collection used many companies less secure java superficial ways several web frameworks web development code brief expressive ajax comet support scalable frameworks lift etc supports pattern matching functions xml support lift support rest web services objectoriented orthogonal complete language java richer types fully reflected bytecode scala reflection library development aspectj really typesafe static typing memory management creating new aspect several crosscutting concern like monitor detect app failure error handling repair authentication verification sessions management complicated development deployment compare used monitor multiple web service service call services failure repair aspect similarities like class supports basic principles like abstraction encapsulation inheritance partial cases powerful controlled mechanism reflection reflection api static typing dynamic typing available declaration garbage collection flexible convenient faster development deployment simple faster development lots enhancement better web primarily language objectoriented paradigm supports reflection namespace comparative studies programming languages within diverse criteria team term report application development haskell thread safe static typing type casting done explicit conversion function module system used access control available internal garbage collection monad transformers libraries used add application specific functionality hxml toolbox using generic data model style achieved class types static reflection using derive tool java robust statically typed implicit type casting access control available subtyping possible automatic garbage collection java api servlet jndi etc java specification requests submitted java community process object oriented static dynamic reflection scheme dynamically strongly typed language implicit type casting access control available subtyping possible automatic garbage collection core language extended libraries allow web app development core language extended libraries allow web services development core language extended libraries allow development style classes objects inheritance polymorphism etc powerful reflection mechanism homoiconic language capable program data treated uniformly dynamic language program expansion comparative studies programming languages within diverse criteria team term report dynamic structure inspection bpel statically strongly typed security requirements part standard java implementations language web app via web clients invoking bpel web services bpel built goal support web services composition bpel language currently support paradigm currently support however ongoing research extend language criteria aspectorientation functional programming declarative programming batch scripting prototype design aspects implement crosscutting concerns modular way feature weave advice code destination code rich library supports functional programming pro mainly used dealing databases allows developer connect database extract manipulate sql statements pro declarative statements mainly used execute sql statements batch scripting done using standard library execute system commands prototyping achieved via libraries api comparative studies programming languages within diverse criteria team term report javascript javascript feature weave advice code destination code javascript support standard functions map reduce select javascript declarative programming achieved using functional library javascript allows execution system command shown javascript used create webbased gui running client side php supported via several extension aop libarary php others performance issues basic attempt bring functional programming php supported programmers adopt declarative style programming code selfdocumented supports scripting batch processing via plugins cron supports via utilizes php object model support platform independent full documentations book scala implementation aop scala supports mixins enable separation code crosscuts class hierarchies supports functional style considered statically typed object oriented language closures supports functional paradigm declarative declarative reading math definition supports real scripting access complete jdk file operations high object oriented level supports based scala library provides access java swing framework gui classes alternative frameworks aspectj specially designed programming direct support extension achieve functional programming achieved base language direct support extension achieve declarative style achieved base language possible monitor automate schedule etc api base language creating aspect certain jointpoints policy rules regulation updates etc notified implementing aop basic designing performed base language java comparative studies programming languages within diverse criteria team term report supported implementation like microsoft functional programming achieved implementation lambda expression namespace supported built library linq regex external commands executed namespace also support internal command automation extensive supports tools control ide considered one best ide design haskell embedding style provides trex pure functional declarative achieved complex way api done portable native library java implementation aspectj mimics functional style pure facility integrated jsetl library process builder class provides facility provides lightweight heavyweight components scheme recent research development extend implementation scheme macros allow specifications advice poitcuts etc supporting aop supported via functions lang constructs allow sideeffect free programming based lambda calculus nonpure functional declarative virtue functional concise expression batch scripts widely used purpose core language extended libraries support development bpel via extension language much ongoing research functional language language constructs imperative structured programming notion primarily imperative language constructs support declarative programming bpel abstract process possible via java language code inserted bpel script done prove concept native support via language allow human interaction workflow comparative studies programming languages within diverse criteria team term report function potential expressed declaratively ongoing research declarative extensions secure programming practices per studies java haskell bpel languages specifically support static typing javasript dynamically typed means checks data types execution time scheme hand dynamically yet strongly typed language according findings scala static strongly typed uses type inference deduce type aspectj observed language researches php dynamically weakly typed language java haskell bpel scheme secure programming languages although provide different manner addition automated garbage collection used safe memory management provided haskell java scheme scala php bpel implementations found built top java use capabilities language web application development languages provide capabilities build web applications php popular according research java builds high quality web applications using embedded apis scala several web frameworks web development scheme programming constructs allow building secure web server web applications extending core language macros haskell relies external libraries building applications extensive supports web development visual studio ide bpel expose wsdl interface browsers invoke services provided aspectj provides better way implement concern without hampering existing code found aspectj functionality exposed web application supports web development hiding many network communications along web browsers javascripts mainly used web applications running script user side help distribute computations reduce network bandwidth web service design composition languages expose functionality via wsdl interfaces however composing web services bpel language built purpose using framework provides good support interoperability web services uses helps web service messaging establishes communication across different platforms javascript uses api web services web service across different platforms comparative studies programming languages within diverse criteria team term report abstraction java scala php languages scheme language supports paradigm aspectj program implementation also supports properties language haskell help type classes use feature object oriented based programming style support class instance method message passing inheritance abstraction encapsulation subtype polymorphism javascript support simple paradigm support inheritance virtual functions polymorphism reflection analysis research found reflection supported languages example java supports static dynamic reflection haskell need separate tool derive avail support reflection scheme homoiconic language reflection supported natively programs language expanded compiled run time php comes complete reflection api reverse engineered classes interfaces functions methods extensions reflection supported namespace construct whereas aspectj offers alternative way access static dynamic context reflection api bpel currently support reflection however ongoing research expand language support reflection mechanism scala support reflection java richer types fully reflected bytecode support compile time reflection support reflection due unrecognizing detailed type information hidden compiler however reflection limited features monitoring expression types querying type name javascript supports reflection object construct provides program reflectedobjects contents programming comparative analysis observed aspectj language designed aop languages require extensions libraries aop support example aspectj one extensions java provide aop bpel uses provide support paradigm haskell achieves aop via ghc compiler approaches facilitate aop php phpaspect php others however extensions performance issues implementation aop scala however scala supports mixins enable separation code crosscuts class hierarchies also supporting extensions aop updated developed one scheme supports aspectorientation via powerful macro system extends core language one extension implement crosscutting concerns modular way supports code pointcuts name pointcuts javascript feature weave advice code destination code helps change behavior program without refractor original code advice code written separate file woven original code execution time functional programming haskell pure functional language side effects contrast scheme language including functional however allows side effects java features comparative studies programming languages within diverse criteria team term report functional programming side effects bpel support functional paradigm according research attempt define lots useful functions php however basic scala programming language supports functional style yet consider really functional language statically typed object oriented language closures also programming language supports functional programming aspectj extension java hand support functional programming paradigm subset java according studies extension language rich library supports functional programming javascript support functional programming map reduce select javascript block scope function scope helps maintain states good functional paradigm declarative programming languages support functional paradigm considered declarative according references include haskell scheme scala however among languages haskell pure functional language php declarative programming language however adopt declarative style programming studies show aspectj bpel support declarative paradigm java libraries support declarative programming cpp imperative object based functional programming paradigm according understanding pro extension cpp language used dealing databases declarative style functional library used javascript achieve declarative programming batch scripting batch scripting supported languages however scheme scala haskell allow expressing scripts concisely found php also supports scripting batch processing via plugins cron hand java support via libraries bpel aspectj allow writing scripts using java library uses dynamic programming compiled compiler considered one script languages javascript allows execution system commands prototype design extensive support prototype design using ide aspectj provide support monitoring user interface threading synchronization notification etc basic designing core concerns achieved base language java php scheme allow creating interfaces via toolkit order support human interaction bpel extended language development gui applications scala provides library access java swing framework gui classes haskell also uses different libraries provide toolkits interface development prototyping achieved via libraries api javascript used create gui running client side comparative studies programming languages within diverse criteria team term report conclusion made comprehensive research survey ten languages support popular paradigms based specified criteria found javascript php aspectj considered safe languages example allows pointers access memory aspectj binding pointcut advice give rise type errors runtime furthermore javascript php weakly dynamically typed languages also considered unsafe hand scala haskell java scheme bpel safe languages among scheme one dynamically typed web application developed using languages studied javascript language specially designed application development among used web application development php java well suited bpel specifically designed standardized language web service composition found languages capable implement web services whereas javascript invoke web services client research done found bpel javascript support objectoriented paradigm constructs inheritance aspectj support basic principles paradigm languages capabilities either natively via libraries scheme java aspectj php powerful reflection mechanism haskell supports static reflection via tool reflection scala reflection library development phase javascript also accomplishes reflection via tools like extendscript aspectj language built specifically javascrpit achieve partial aop feature weave advice code destination code languages aop properties via libraries extensions javascript php obtain functional style via either libraries extensions hand scheme haskell scala considered functional languages among haskell one pure functional implementation functional programming observed aspecj base language java mimic functional styles bpel support functional paradigm research ongoing achieve functional languages considered declarative definition although languages researched support batch scripting accomplish concise way aspectj bpel also support batch scripting using java api base language support gui provided javascript uses asp web form server side controls built web gui support ide additionally java applets good example applications running browsers bpel allows human interaction workflow via another language called scheme php java servlet technology create html web pages sent http client displayed browser studied languages support command prompt prototyping non gui also supported via libraries toolkits addition base research work also studied popularity programming languages according tiobe programming community index august http comparative studies programming languages within diverse criteria team term report found scheme increased popularity compared august hand java php remain position last year addition according source ranking popular languages least follows java php scheme haskell scala per tiobe categories programming languages statically typed languages popular four years fact languages increased popularity since last year functional logical programming languages rank procedural languages however increased popularity according http following rankings powell books google code ohloh slashdot placed java top position moreover rankings like yahoo search results discussion site results lambda ultimate show ranks first references lloyd advantages declarative 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team term report appendix installing racket scheme dialect used paper testing racket formally known plt scheme downloaded http racket run command line racket graphical environment drracket programs related writing web applications need saved directory racket installed http test installation create small program text editor drracket like following save program run command prompt using commands enter loading files enter command see output web server scheme following implementation server scheme ran racket taken whole commented also slightly modified produce sum number entered comparative studies programming languages within diverse criteria team term report figure startup server accept port user running scheme server start server first open racket command prompt enter enter new prompt enter define stop serve server running open browser window enter http web page ready accept number displayed enter number press submit button response page displayed many number entered code fragment figure define serve function listens client requests invokes listener function thread process request comparative studies programming languages within diverse criteria team term report figure listener function definition function invokes handle function passes input output streams figure function invokes define dispatch function path string figure figure handle function definition comparative studies programming languages within diverse criteria team term report figure dispatch function definition dispatch table declared shown figure map string path function handler implemented hash data structure handler found error response generated client otherwise appropriate handler function retrieved dispatch table invoked comparative studies programming languages within diverse criteria team term report figure define many query define reply query label hidden function definitions define many query invoked handle request calling helper function label hidden figure build request form sends form another user responds handler retrieved dispatch table invoked builds html page hello corresponding number received alternative response commented bottom code snippet adds number displays sum browser appropriate message run code changes rest program need done part original code comparative studies programming languages within diverse criteria team term report figure building dispatch table built using scheme imperative special form key value function handles figure request form displayed browser window figure response produced server request form displayed figure response figure web server bpel similar behavior achieved using bpel language sine primary goal bpel support automated interactions web services communicating using soap another transport protocol http xml wire protocol data exchange protocol standard interfaces basic structure basic invocation differs one scheme example input output represented using xml document conforms xml grammar rules structure data defined xsd schema validates data scheme data passed server request sent browser invoke correct functionality scheme uses bpel requests wired directly control statements handle request bpel seen single function possibly many nested scopes event message handlers exception handlers correlations grouped together part computation performed within bpel module part delegated web services invoking interfaces base structure functionality implements simple web application obtained existing synchronoussample comes netbeans online tutorial obtain use found online http comparative studies programming languages within diverse criteria team term report document describes interface bpel defining structure input output messages data remain unmodified data consists single type input output fields wrapped messages described wsdl entirely modified implement functionality application implemented using primarily design view automatically generates equivalent bpel code visual model design view palette obtain programming constructs bpel supports application consists nested sequence another sequence scopes first assignment within outer sequence scope used initialize output predicate expression scope compares input zero zero control enters scope within true path scope nested sequence within two assignments one used decrement input used later predicate concatenating string output output wrapped xml document test cases generated using integrated junit included netbeans ide input output one test case generated source code seen figure figure figure respectively figure graphical representation server application implemented using bpel comparative studies programming languages within diverse criteria team term report figure sample input represented xml figure sample output represented xml comparative studies programming languages within diverse criteria team term report figure code fragment implementing functionality generate bpel engine factorial function implemented bpel generated source code test input corresponding output shown figure figure figure comparative studies programming languages within diverse criteria team term report figure generated source code factorial function figure input test factorial function implemented bpel comparative studies programming languages within diverse criteria team term report figure output testing factorial function implemented bpel input php creating web service create basic web service provides xml json response using php mysql author david walsh php mysql require user parameter isset intval soak passed variable set isset intval default format strtolower default intval default connect link die connect link die select grab posts query select guid order desc limit result query link die query query create one master array records posts array result post result posts array post output necessary format format header echo array posts else comparative studies programming languages within diverse criteria team term report header echo posts foreach posts index post post foreach post key value echo key value foreach value tag val echo tag htmlentities val tag echo key echo link disconnect take following sample url example http xml output posts post sslmatic ssl certificate giveaway winners guid http post mootools filemanager guid http post phptvdb using php retrieve show information guid http post david walsh lost mootools plugins guid http post create short urls using guid http post create short urls using php guid http post represent repositories using github badge guid http post zebratable guid http comparative studies programming languages within diverse criteria team term report post mootools zebra table plugin guid http post sslmatic quality cheap ssl certificates giveaway guid http json output posts post sslmatic ssl certificate giveaway winners guid http post mootools filemanager guid http post phptvdb using php retrieve show information guid http post david walsh lost mootools plugins guid http post create short urls using guid http post create short urls using php guid http post represent repositories using github badge guid http post zebratable guid http post mootools zebra table plugin guid http post sslmatic quality cheap ssl certificates giveaway guid http scala building web services lift build web services lift author steve jenson start simple class package import import import class blogapi val request requeststate extends simplecontroller def index responseit filled def get string responseit filled comparative studies programming languages within diverse criteria team term report def create responseit filled def delete string responseit filled get existing item get route request proper method via pattern matching def index responseit request match case requeststate parsepath api itemid nil getrequest get itemid def get string responseit val user match case full item atomresponse case empty notfoundresponse outputting xml makes really easy output valid xml following item model object def toatom val http val formatter new simpledateformat val updated entry http updated updated author name name content xhtml div http body comparative studies programming languages within diverse criteria team term report php abstract class basics following example creating simple abstract class extends successfully extend oophpabstractclass oophpclasstoextendanabstract must getname setname functions lawrence truett rights reserved class extends oophpabstractclass class oophpclasstoextendanabstract extends oophpabstractclass private instancename abstract function getname must implement public function getname return instancename abstract function setname must implement public function setname namein instancename namein lawrence truett rights reserved simple php abstract class defines two functions getname setname namein class extending must abstract class oophpabstractclass abstract public function getname abstract public function setname namein comparative studies programming languages within diverse criteria team term report lawrence truett rights reserved define echo testing php abstract echo echo create class extends classone new oophpclasstoextendanabstract echo setname harold echo getname echo echo testing php abstract output begin testing php abstract classes test create class extends abstract harold end testing php abstract classes php reflection reflection example shell terminal php strlen php finfo php json php dom example output something similar function internal core function strlen parameters comparative studies programming languages within diverse criteria team term report parameter required str class internal fileinfo class finfo constants static properties static methods properties methods method internal fileinfo ctor public method finfo parameters parameter optional options parameter optional arg method internal fileinfo public method parameters parameter required options method internal fileinfo public method file parameters parameter required filename parameter optional options parameter optional context method internal fileinfo public method buffer parameters parameter required string parameter optional options parameter optional context extension persistent extension json version constants comparative studies programming languages within diverse criteria team term report constant integer constant integer constant integer constant integer constant integer constant integer constant integer constant integer constant integer constant integer functions function internal json function parameters parameter required value parameter optional options function internal json function parameters parameter required json parameter optional assoc parameter optional depth function internal json function parameters dom enabled api version libxml version html support enabled xpath support enabled xpointer support enabled schema support enabled relaxng support enabled
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aug bass numbers local rings via stable cohomology luchezar avramov srikanth iyengar abstract finite module finite projective dimension noetherian local ring maximal ideal residue field proved natural map extr extr regular zero otherwise noteworthy aspect proof use stable cohomology applications include computations bass series certain local rings introduction let denote commutative noetherian local ring maximal ideal residue field regular say singular article revolves around following result theorem singular local ring finite projective dimension extr canonical map special cases known long time surveyed end section even cases proof new utilizes result martsinkovsky properties vogel stable cohomology functors recalled section also suggests extensions modules certain commutative algebras see applications theorem include new criteria regularity local rings section explicit computations bass numbers modules section stable cohomology section recall construction stable cohomology basic results required sequel approach adopt based construction vogel described goichot see also let associative ring let denote center given left choose projective resolutions respectively recall homomorphism degree family maps element homr homr module component complex homr differential date version august mathematics subject classification primary secondary key words phrases bass numbers stable cohomology fiber products research partly supported nsf grants lla sbi avramov iyengar maps form subcomplex component homr homr quotient complex independent choices write hom homotopy exact sequence complexes homr homr hom stable cohomology pair graded ext hom ext comes equipped functorial homomorphisms graded extnr ext pdr pdr finite ext indeed case may choose bounded complex definitions yield homr homr hence hom family every integer canonical inclusions induce functoriality commutative diagram extnr extr ext ext proposition suppose admits resolution finite projective every integer vertical maps bijective particular map injective surjective corresponding property every proof let resolution finite projective projective resolution complex projective resolution commutative diagram morphisms complexes homr homr hom homr homr hom natural vertical maps map bijective represents extnr extnr bijective due hypothesis evidently bijective isomorphism square diagram induces bass number local rings local rings next theorem main result paper concerns maps extnr extnr extnr induced homomorphism derived result martsinkovsky using properties stable cohomology recalled theorem let local ring singular map factors module finite projective dimension extnr proof hypothesis factors finite projective dimension following diagram extn extnr extnr ppp ppp ppp extn ppp extn extnr ext ext ext commutes due naturality maps involved equality comes map injective theorem proposition shows injective well diagram yields extnr note finiteness condition imposed theorem remark used proof following corollary deals maps torr torn torn induced homomorphism corollary singular map factors module finite injective dimension torr proof set homr injective envelope let factorization finite injective dimension ishikawa module finite flat dimension finite projective dimension jensen factors theorem gives extnr natural isomorphism extnr torr yields torn whence get torn desired next record elementary observation homr lemma let local ring map coker free direct summand ker holds free finite rank converse assertion holds well proof condition coker holds epimorphism coker map exists ker otherwise finite free ker holds pick ker since finite free extended basis hence avramov iyengar theorem introduction crucial implication next result theorem let local ring let extnr extnr extnr map induced natural map following conditions equivalent regular integer iii pdr integer every finite dim coker minimal free resolution free direct summand integer proof set homr homr minimality get commutative diagram extnr extn regular take koszul complex minimal generating set gives isomorphism inequality nakayama lemma diagram yields iii implications tautologies iii implication special case theorem preceding diagram shows condition equivag lent ker thus desired assertion follow lemma notes equivalence conditions theorem proved ivanov theorem gorenstein lescot general equivalence due dutta shown follows lescot theorem via elementary lemma martsinkovsky deduced dutta theorem theorem used latter prove regularity criteria different iii theorem see bass numbers modules nth bass number module local ring integer rankk extnr follows given homomorphism let denote submodule theorem let local ring map set rankk singular pdr finite equality bass number local rings proof set let induced maps appear commutative diagram exact row since split get commutative diagram exact row extr ext extr extr extr extr zero map due theorem implies extr definition exists exact sequence extr cohomology sequence yields exact sequence extr extnr extnr spaces integer computing ranks using morphism extr extr obtain desired equality recall nth betti number integer rankk extnr corollary assume singular holds set rankk finite pdr holds equality proof hypotheses give rankk applying theorem submodules obtain rankk pdr finite foxby yields remark hypothesis pdr corollary necessary otherwise sum defined hand pdr necessarily conclusion corollary may fail example finite free rank dim indeed follows cohomology exact sequence induced exact sequence extnr map bijective proof theorem avramov iyengar bass numbers often described terms rgenerating formal power series also use series terms formulas restated equality pnr stpkr let local rings let denote canonical maps fiber product defined formula well known easy see subring local maximal ideal residue field set let finite modules respectively canonical maps turn lescot proved holds fiber product natural structure finite corollary notation set rankk singular pds pdt finite iss pns itt ppt pkr pks pkt proof whence first equality ppt iss pns pkt second one comes applying formulas order notes corollaries specialize lescot results respectively proof presented corollary faithfully transposes derivation finite submodule containing proved theorem bass numbers asymptotically size measured appropriate polynomial exponential scales closed formula corollary much precise statement noted remark hold pdr infinite regular references avramov modules extremal resolutions math res letters avramov iyengar evaluation maps stable cohomology algebras preparation avramov stable cohomology local rings adv math dutta syzygies homological conjectures commutative algebra berkeley math sci res inst publ springer new york bass number local rings foxby isomorphisms complexes applications homological theory modules math scand goichot homologie pure appl algebra ishikawa injective modules flat modules math soc japan ivanov homological characterization certain class local rings math ussr sbornik jensen les foncteurs lim leurs applications des modules lecture notes math springer berlin lescot bass produit anneaux locaux dubreilmalliavin paris lecture notes math springer berlin martsinkovsky remarkable property syzygy modules residue field nonregular local ring pure appl algebra department mathematics university nebraska lincoln address avramov department mathematics university nebraska lincoln address
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manuscript version made available license http preprint version final version paper available http particle swarm optimization generating interpretable fuzzy reinforcement learning policies daniel heina alexander hentschelc thomas runklerb steffen udluftb aug technische department informatics boltzmannstr garching germany corporate technology munich germany axiomzen howe vancouver canada siemens abstract fuzzy controllers efficient interpretable system controllers continuous state action spaces date controllers constructed manually trained automatically either using cost functions incorporating detailed knowledge optimal control strategy requirements automatic training processes found reinforcement learning problems applications online learning often prohibited safety reasons requires exploration problem dynamics policy training introduce fuzzy particle swarm reinforcement learning fpsrl approach construct fuzzy policies solely training parameters world models simulate real system dynamics world models created employing autonomous machine learning technique uses previously generated transition samples real system best knowledge approach first relate fuzzy controllers batch fpsrl intended solve problems domains online learning prohibited system dynamics relatively easy model previously generated default policy transition samples expected relatively easily interpretable control policy exists efficiency proposed approach problems domains demonstrated using three standard benchmarks mountain car balancing experimental results demonstrate interpretable fuzzy policies keywords interpretable reinforcement learning fuzzy policy fuzzy controller particle swarm optimization introduction work motivated typical industrial application scenarios complex industrial plants like wind gas turbines already operated field years plants control realized dedicated controllers guarantee safety stability controllers constructed respect plant subsystem dependencies modeled expert knowledge complex mathematical abstractions first principle models finite element methods examples controllers include fuzzy controllers considered efficient interpretable casillas cordon herrera corresponding author email address daniel hein preprint submitted engineering applications artificial intelligence magdalena system controllers control theory decades procyk mamdani scharf mandve shao however observed control usually implemented default control strategies provided best practice approaches domain experts maintaining system based personal experience knowledge system dynamics one reason lack autonomously generated controllers modeling system dependencies control dedicated mathematical representations complicated often infeasible approach modeling representations differentiable equations required classical controller design even complicated since many applications equations found training controllers performed reward samples august plant reinforcement learning sutton barto capable yielding controllers based solely available system data generally concerned optimization policy system modeled markov decision process policy maps system states actions system repeatedly applying policy generates trajectory space section learning controllers way produces interpretable controllers scope paper proposed approach especially industry problems high interest since interpretable policies expected yield higher acceptance domain experts solutions maes fonteneau wehenkel ernst fundamental difference classical control theory machine learning approaches lies way techniques address stability reward function design classical control theory stability central property controller example lyapunov stability theory analyzes stability solution near point equilibrium widely used design controllers nonlinear systems lam zhou moreover fault detection robustness interest fuzzy systems yang liu hua yang liu zhao yang shi problems addressed classical fuzzy control theory stability fault detection robustness make well suited serving system controllers controllers reward functions specifically designed purpose parameter training essential contrast second view defining reward functions typically applied system control sample system latent underlying reward dynamic subsequently use data perform machine learning herein apply second view defining reward functions capable utilizing sampled reward data controller training note goal find policy maximizes trajectory expected accumulated rewards referred return value without explicitly considering stability several approaches autonomous training fuzzy controllers proven produce remarkable results wide range problems jang introduced anfis fuzzy inference system implemented using adaptive network framework approach frequently applied develop fuzzy controllers example anfis successfully applied balancing problem saifizul azlan mohd nasir hanafy kharola gupta anfis training process training data must represent desired controller behavior makes process supervised machine learning approach however optimal controller trajectories unknown many industry applications feng applied particle swarm optimization pso generate fuzzy systems balance system approximate nonlinear function debnath optimized gaussian membership function parameters nonlinear problems showed parameter tuning much easier pso conventional methods knowledge derivative complex mathematical equations required debnath shill murase kothandaraman ponnusamy applied pso tune adaptive neuro fuzzy controllers vehicle suspension system however similar anfis pso fitness functions contributions dedicated expert formulas error functions depend correctly classified samples best knowledge fuzzy rules never combined batch approach proposed fuzzy particle swarm reinforcement learning fpsrl approach different fuzzy policy parameterizations evaluated testing resulting policy world model using monte carlo method sutton barto combined return value number action sequences fitness value maximized iteratively optimizer batch consider applications online learning approaches classical learning sutton prohibited safety reasons since approaches require exploration system dynamics contrast batch algorithms generate policy based existing data deploy policy system training setting either value function system dynamics trained using historic operational data comprising set form observation action reward next observation referred data batch research past two decades gordon ormoneit sen lagoudakis parr ernst geurts wehenkel suggests batch algorithms satisfy system requirements particularly involving neural networks nns modeling either value function riedmiller schneegass udluft martinetz riedmiller gabel hafner lange system dynamics bakker depeweg doshivelez udluft moreover batch algorithms riedmiller udluft zimmermann batch data utilized repeatedly training phase fpsrl approach training conducted environment approximation referred world model generating world model real system data advance training fuzzy policy offline using model several advantages many scenarios data describing system dynamics available advance easily collected policies evaluated real system thereby avoiding detrimental effects executing bad policy reward function engineering yielding differentiable equation utilized policy training required sufficient sample system reward function model underlying dependencies using supervised machine learning remainder paper organized follows methods employed framework reviewed sections specifically problem finding policies via formalized optimization task addition review membership functions describe proposed parameterization approach finally pso optimization heuristic use searching optimal policy parameters different extensions presented overview proposed fpsrl approach derived different methods given section experiments using three benchmark problems mountain car problem balancing cpb task complex cpsu challenge described section section also explain setup process world models introduce applied fuzzy policies experimental results discussed section results demonstrate proposed fpsrl approach solve benchmark problems understandable benchmark fpsrl compare obtained results neural fitted iteration nfq riedmiller established technique note technique chosen describe advantages limitations proposed method compared widely available standard algorithm action strategy policy maximizes expected cumulative reward return proposed setup goal find best policy among set policies spanned parameter vector herein policy corresponding particular parameter value denoted state policy outputs action policy performance starting measured return accumulated future rewards obtained executing policy account increasing uncertainties accumulating future rewards reward future time steps weighted furthermore adopting common approach include finite number future rewards return sutton barto expressed follows discount factor selected end time horizon last reward accounted weighted yielding overall policy performance obtained averaging starting states using respective probabilities wst weight factors thus optimal solutions problem wst arg max optimization terminology policy performance function referred fitness function many problems cost executing potentially bad policy prohibitive pilots learn flying using flight simulator instead real aircraft similarly busoniu babuska shutter ernst state transition function approximated using model first principle model created previously gathered data substituting place state transition function obtain approximation true fitness function study employ models based nns however proposed method extended models bayesian nns depeweg gaussian process models rasmussen williams reinforcement learning biological learning animal interacts environment attempts find action strategies maximize perceived accumulated reward notion formalized area machine learning acting agent explicitly told actions implement instead agent must learn best action strategy observed environment responses agent actions common challenging problems action affects next reward subsequent rewards sutton barto examples delayed effects nonlinear change position force applied body mass delayed heating combustion engine formalism agent interacts target system discrete time steps time step agent observes system state applies action state space action space depending system transitions new state agent receives reward herein focus deterministic systems state transition reward expressed functions respectively desired solution problem fuzzy rules fuzzy set theory introduced zadeh based theory mamdani assilian introduced fuzzy controller specified set linguistic rules whose membership functions activated independently produce combined output computed suitable defuzzification function system rules fuzzy rule expressed follows iteration particle remembers best local position visited far including current position furthermore particle knows neighborhood best position denotes input vector environment state setting membership fuzzy set input vector premise part real number consequent part paper apply gaussian membership functions wang mendel popular type membership function yields smooth outputs local never produces zero activation forms multivariate gaussian function applying product membership dimensions define membership function rule follows exp parameterized gaussian center width parameter vector set valid gaussian fuzzy parameterizations size contains xmin xmax distributed uniformly velocity vector contains cognitive component social component represent attraction given particle best position neighborhood best position respectively velocity vector calculated follows vij yij xij found far one particle neighborhood including neighborhood relations particles determined swarm population topology generally fixed irrespective particles positions note ring topology eberhart simpson dobbins used experiments described section let denote position particle iteration changing position particle iteration achieved adding velocity vector particles position vector arg max cognitive component xij social component output determined using following formula tanh inertia weight factor vij xij velocity position particle dimension positive acceleration constants used scale contribution cognitive social components yij respectively factors random values sampled uniform distribution introduce stochastic element algorithm best position particle maximization problem iteration calculated follows else hyperbolic tangent limits output parameter used change slope function particle swarm optimization pso algorithm stochastic optimization heuristic generally pso operate search space bounded vector space kennedy eberhart position particle swarm represents potential solution given problem particles fly iteratively multidimensional search space referred fitness landscape movement particle receives fitness value new position fitness value used update particle velocity vector velocity vectors particles certain neighborhood framework fitness function given particle positions represent policy parameters pseudocode pso algorithm applied experiments section provided appendix fuzzy particle swarm reinforcement learning basis proposed fpsrl approach data set contains state transition samples gathered real system samples represented tuples state action applied resulted state transition yielded reward generated using even random policy prior policy training generate world models inputs predict using data set advantageous learn differences state variables train single model per state variable separately yield better approximative quality alternative modeling technique use regression trees breiman friedman olshen stone typically lacking data efficiency regression tree predictions less affected nonlinearities perceived system dynamics rely functional approximation employed pso study optimizer require gradient information fitness landscape pso utilizes neighborhood information systematically search valuable regions search space note gradientdescent based methods evolutionary algorithms alternative techniques experiments mountain car benchmark underpowered car must driven top hill fig achieved building sufficient potential energy first driving direction opposite final direction system fully described state space representing cars position velocity conducted experiments using freely available cls software clsquare benchmark system applies method approximate closed loop dynamics task agent find sequence force actions drive car hill achieved reaching position start episode car position initialized interval agent receives reward otherwise resulting state calculated according reward also given thus reward function approximated using next fpsrl step assumption number rules per policy required experiments started minimal rule set benchmark calculated respective performances increased number rules compared performance policies fewer rules process repeated performance respect dynamic models satisfactory intuitive representation maximal achievable policy performance given certain discount factor respect particular model computed adopting trajectory optimization technique hein hentschel runkler udluft prior fpsrl training optimization particle position pso represents parameterization fuzzy policy fitness particle calculated generating trajectories using world model starting fixed set initial benchmark states section schematic representation proposed fpsrl framework given fig note present results fpsrl using nns world models pso optimization technique considered problem domain continuous smooth deterministic system dynamics nns known serve adequate world models given batch previously generated transition samples training process training errors excellent indicators well model perform training nevertheless different problem domains alternative types world models might preferable example gaussian processes rasmussen williams provide good approximation mean target value technique indicates level confidence prediction feature may value stochastic system dynamics second subsequent state transition car reaches goal position position becomes fixed velocity becomes zero agent perceives maximum reward following time step regardless applied actions balancing experiments described following two sections also conducted using cls software objective cpb benchmark apply forces cart moving track keep pole hinged cart upright position fig four markov state variables pole angle pole angular velocity cart position cart velocity variables describe markov state completely additional information system past behavior required task agent find sequence force actions prevent pole falling fantoni lozano http gradient descent regression trees evolutionary algorithms gaussian processes pso fuzzy rules neural networks figure schematic visualization proposed fpsrl approach left right pso evaluates parameter vectors predefined fuzzy rule representation given set parameters evaluation performed first computing action vector state approximative performance tuple computed predicting resulting state transition reward using nns repeating procedure state successor states generates approximative trajectory state space accumulating rewards using return computed state eventually used compute fitness value drives swarm high performance policy parameterizations alternative techniques could replace pso nns presented background figure mountain car benchmark task first build momentum driving left order subsequently reach top hill right cpb task angle pole cart position restricted intervals respectively cart left restricted area episode considered failure velocities become zero cart position pole angle become fixed system remains failure state rest episode policy apply force actions cart time intervals reward function balancing problem given follows otherwise figure benchmark task balance pole around moving cart position applying positive negative force cart keep rest episode since problem symmetric around optimal action state corresponds optimal action state thus parameter search process simplified necessary search optimal parameters one half fuzzy policy rules half parameter sets constructed negating membership functions mean parameters respective output values policy components note membership function span width fuzzy rules parameter negated membership functions must preserve shapes based reward function primary goal policy avoid reaching failure state secondary goal drive system goal state region approximation next state given following formula cpsu benchmark based system dynamics cpb benchmark contrast cpb benchmark position cart angle pole restricted consequently pole swing important property cpsu since pole angle initialized full interval often necessary policy swing pole several times side side gain sufficient energy erect pole receive highest reward cpsu setting policy apply actions cart reward function problem given follows otherwise result formula subsequently used approximate state transition reward training sets benchmarks samples originate trajectories cpb cpsu state transitions generated random walk benchmark dynamics start states trajectories sampled uniformly cpb cpsu conducted several experiments investigate effect different data set sizes different network complexities results give detailed impression minimum amount data required successfully apply proposed fpsrl approach different benchmarks adequate complexity data batch size experiments conducted network complexities one two three hidden layers hidden neurons arctangent activation functions training used algorithm neuneier zimmermann training networks executed parallel requires couple minutes detailed overview approximation performance resulting models fpsrl rules created models comparison policies generated nfq data sets given tables mean squared errors normalized output variables standard depicted respect generalization data sets similar reward function cpb benchmark contain penalty failure states terminate episode reached neural network world models conducted policy training world models yielded approximative fitness functions section experiments created one state variable prior training respective data sets split blocks training validation generalization sets respectively weight updates training computed utilizing training sets weights performed best given validation sets used training results finally weights evaluated using generalization sets rate overall approximation quality unseen data nns trained data set dmc containing tuples trajectories generated applying random actions benchmark dynamics start states trajectories uniformly sampled random position track zero velocity following three nns trained approximate task policy representations proposed fpsrl approach search parameterization fuzzy policy formed certain number rules performance fpsrl policy related number rules rules generally allow sophisticated reaction system states hand higher number rules requires parameters optimized makes optimizer search problem difficult addition complex set rules tends difficult even impossible interpret thus determined two rules sufficient cpb benchmarks adequate performance cpsu benchmark achievable minimum four rules output fpsrl policies continuous although output obtained increasing parameter compared fpsrl policy training performance applying nfq problems using data sets approximative models nfq chosen widely applied similarly dynamic model state created following four networks methodology used nfq implementation teachingbox tool box paper aim claim proposed fpsrl approach superior best algorithms terms performance thus nfq selected show degree difficulty benchmarks advantages limitations proposed method recent developments deep produced remarkable results online benchmarks silver lever heess degris wierstra riedmiller van hasselt guez silver future studies may reveal performance offline problems superior nfq nevertheless methods attempt produce interpretable policies drive car hill initial state less time steps one way visualize fuzzy policies plot respective membership functions analyze produced output sample states graphical representation policy benchmark given fig time consideration able understand policy outputs considered state balancing cpb benchmark two different discontinuities dynamics make modeling process difficult compared benchmark case first discontinuity occurs cart leaves restricted state space ends failure state soon cart becomes fixed current position velocities become zero second discontinuity appears cart enters goal region region reward switches rather small change compared failure state reward addition difficulty modeling discrete changes nns task becomes even complicated samples transitions rare contrast difficulties modeling benchmark dynamics rather simple policy balance pole without leaving restricted state space discount factor consider policies yield performance greater successful task fpsrl find parameterization two fuzzy rules used particles pso setup appendix training employed start states uniformly sampled table note data batch size sample transitions required build models adequate approximation quality training policy models trained sample transitions could correctly approximate effects occurred entering area incorrectly predicted possibilities escape failure state balance pole subsequent time steps fpsrl technique exploited weaknesses produced policies perform well models demonstrated poor performance real benchmark dynamics visual representation one resulting fuzzy policies given fig example illustrates situation potential problems policy observed via visual inspection significant advantage interpretable policies contrast fpsrl nfq could produce wellperforming policies even small data batch sizes note weak models used fpsrl training used determine nfq iteration produced best policy nfq training episodes experiments observed even models bad approximative quality results mountain car performed nfq training procedures benchmark using setup described appendix nfq iteration latest policy tested world model compute approximation real performance policy yielding best fitness value thus far saved intermediate solution evaluate true performance nfq policies computed true fitness value applying policies mathematical dynamics difficulties benchmark discontinuity velocity dimension reaching goal rather long horizon required observe effects applied actions first problem difficult model goal area condition limited samples reaching goal using random policy training errors lead situation models correctly represent state transitions velocity suddenly becomes rather models learn yields subsequently fpsrl training evaluation policy candidates results situation car driven kept area applying correct forces leads high reward transitions problem could solved incorporating external knowledge goal area would result convenient training process explicitly want incorporate expert knowledge benchmarks instead wanted demonstrate purely autonomous learning example results given table show despite difficulties even small data batch sizes policies learned using fpsrl nfq benchmark discount factor resulting consider policy performance greater successful solution benchmark problem policy performance freely available https rule slope right accelerate velocity rule slope right example norm norm tanh norm figure fuzzy rules benchmark membership functions plotted blue example state position plotted red gray area respective activation grade rules activations maximal nearly position implies minor influence policy output observation fits fact benchmark simplistic policy exists accelerate car direction current velocity although trivial policy yields good performance problem better solutions exist example stop driving left earlier certain position reach goal fewer time steps yields higher average return depicted policy implements advantageous solution shown example section state data models batch size policies layer layers layers fpsrl nfq selected mean std selected mean std selected mean std table results left right data number state transitions obtained random trajectories benchmark dynamics models generalization errors best models able produce given certain amount data network complexity policies performance real benchmark dynamics different policy types according performance using models left policy setting training experiments performed obtain statistically meaningful results presented results different data batch sizes show benchmark dynamics rather easy model using nns addition models significantly greater errors generalization sets still sufficient training fuzzy policy using fpsrl selecting policy nfq simulating benchmark dynamics useful nfq policy selection nfq training never observed models training therefore prone exploiting weaknesses expected compact interpretable proposed fpsrl approach might interest industry domain experts experimental results obtained three standard benchmarks demonstrated advantages limitations proposed method compared nfq approach however results obtained cpb problem reveal important limitation fpsrl training using weak approximation models proposed approach exploit weaknesses potentially result poor performance evaluated using real dynamics modeling techniques provide measure uncertainty predictions gaussian processes bayesian nns possibly overcome problems recent developments modeling stochastic dynamic systems depeweg may provide approximation mean next system state also compute uncertainty transitions space addition continuous state action spaces well long time horizons appear introduce obstacles training fuzzy policies resulting policies obtained cpsu benchmark performed significantly better generated standard nfq however one significant advantages proposed method methods fact fuzzy rules easily conveniently visualized interpreted suggested compact informative approach present fuzzy rule policies serve basis discussion domain experts application proposed fpsrl approach industry settings could prove significant interest many cases data systems readily available interpretable fuzzy policies favored solutions modelfree approaches compared cpb benchmarks results cpsu benchmark show completely different picture terms performance training process despite cpb cpsu sharing underlying mathematical transition dynamics differ following two important aspects first discontinuities state transitions occur owing absence failure state area second planning horizon successful policy significantly higher latter makes particularly difficult find solution applying standard nfq first property makes cpsu good example strength proposed fpsrl approach nfq performance decreased dramatically cpsu problem table benchmark solutions greater set benchmark states uniformly sampled considered successful policies exhibiting performance given test states experiments none nfq trainings produced successful policy contrast proposed fpsrl could find parameterization successful policies using four fuzzy rules assessing performance world models trained data batch sizes greater data batch size transition samples containing goal area reward far data set model area correctly however extremely high errors obtained generalization set model training excellent indicators weakness figure shows even complex fuzzy policies visualized help make policies interpretable acknowledgment project report based supported funds german federal ministry education research project number sole responsibility report contents lies authors authors would like thank dragan obradovic clemens otte insightful discussions helpful suggestions conclusion traditional way create fuzzy controllers either requires fitness function according optimizer finds optimal controller parameters relies detailed knowledge regarding optimal controller policy either requirement difficult satisfy dealing industrial problems however data gathered system controlled using default policy available many cases fpsrl approach proposed herein use data produce interpretable fuzzy policies problems particularly problems system dynamics rather easy model adequate amount data resulting policy references bakker state mind reinforcement learning recurrent neural networks thesis leiden university netherlands breiman friedman olshen stone classification regression trees crc press boca raton busoniu babuska shutter ernst reinforcement learning dynamic programming using function approximation crc press rule accelerate velocity rule example norm norm tanh norm figure fuzzy rules cpb benchmark visualization fuzzy policy useful revealing weaknesses given set rules example despite presented policy result successful fpsrl training yielding high average returns test states states boundary allowed cart position would result failed episodes depicted example shows state would activate rule thus would accelerate car even positive eventually end failure state therefore examining rule set elementary weaknesses policy identified test state set adapted appropriately data models batch size policies layer layers layers fpsrl nfq selected mean std selected mean std selected mean std table balancing results experiments show modeling process variables containing nonlinearities difficult requires adequate amount sample data since pendulum cart velocities suddenly become zero failure state reached modeling process requires certain number events training data correctly model effect results batch size show model applicable still used policy selection technique nfq models errors reduce increasing data batch sizes fpsrl becomes increasingly capable finding interpretable policies note encountered effect reduced nfq performance occur even data batch size increases rule accelerate velocity rule right rule left rule example norm norm tanh norm figure fuzzy rules cpsu benchmark even four rules fuzzy policies visualized easily interpretable way inspecting prototype diagrams rule two basic concepts identified accelerating direction first rule cart position left right moving left right pole simultaneously falling right left cart accelerated towards right left second rule cart center right left pole hanging cart accelerated towards right left prototypes utilized realize complex task swinging pole balancing pole cart centered around realized via fuzzy interaction prototype rules shown example last row data models batch size policies layer layers layers fpsrl nfq selected mean std selected mean std selected mean std table results high errors generalization set training reward function data batch size less clearly indicate absence adequate number transition samples describe effects noted reaching goal area smooth easy dynamics dimensions make rather easy model cpsu dynamics subsequently use conduct note long planning time horizon required benchmark made impossible learn successful policies standard nfq ieee transactions systems man cybernetics kennedy eberhart particle swarm optimization proceedings ieee international joint conference neural networks kharola gupta stabilization inverted pendulum using hybrid adaptive neuro fuzzy anfis controller engineering science letters kothandaraman ponnusamy pso tuned adaptive controller vehicle suspension systems journal advances information technology lagoudakis parr policy iteration journal machine learning research lam zhou dynamic output feedback control fuzzy systems lyapunov function approach international journal systems science maes fonteneau wehenkel ernst policy search space simple formulas towards interpretability reinforcement learning discovery science mamdani assilian experiment linguistic synthesis fuzzy logic controller international journal manmachine studies neuneier 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controller application dynamic processes fuzzy sets systems silver lever heess degris wierstra riedmiller deterministic policy gradient algorithms proceedings international conference international conference machine learning volume icml sutton learning predict methods temporal differences machine learning sutton barto reinforcement learning introduction bradford book van hasselt guez silver deep reinforcement learning double aaai conference artificial intelligence aaai wang mendel fuzzy basis functions universal approximation orthogonal learning ieee transactions neural networks yang liu hua fault detection uncertain fuzzy systems based delta operator approach circuits systems signal processing yang liu zhao robust control nonlinear systems subject actuator saturation via delta operator approach information sciences yang shi control class fuzzy systems via delta operator approach signal process zadeh fuzzy sets information control data randomly initialized particle positions velocities particle fitness function inertia weight factor acceleration constants random number generator rand search space boundaries xmin xmax velocity boundaries vmin xmax xmin vmax xmax xmin swarm topology graph defining neighborhood result global best position repeat foreach particle neighborhood best position particle arg end position updates foreach particle determine new velocity particle vij wvij rand yij xij rand xij end truncate particle velocity vij min vmaxj max vminj vij end compute new position particle truncate particle position xij min xmaxj max xminj xij end personal best positions set new personal best position particle end end stopping criterion met determine global best position arg return appendix pso algorithm algorithm explains pseudocode pso algorithm applied experiments appendix experimental setup algorithm pso algorithm particle represented position personal best position neighborhood best position table gives compact overview parameters used experiments presented herein note extensive parameter studies fpsrl nfq beyond scope paper nevertheless evaluated various parameters known literature cpb cpsu benchmark state dimensionality time horizon discount factor fpsrl number particles pso iterations pso topology number rules rule parameters actions ring ring ring sigmoid sigmoid sigmoid nfq iterations epochs layers activation actions table experimental setup experience noted presented setup successful setups tested experiments
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oct distributed receding horizon control methodology consensus problems fei suna kamran turkoglua aerospace engineering san state university san jose abstract work investigates consensus problem nonlinear systems distributed nonlinear receding horizon control methodology work develop scheme reach consensus nonlinear multi agent systems fixed graph without need linearization techniques purpose problem consensus converted optimization problem directly solved backwards sweep riccati method generate control protocol results algorithm stability analysis conducted provide convergence guarantees proposed scheme addition extension consensus nonlinear systems presented several examples provided validate demonstrate effectiveness presented scheme corresponding theoretical results keywords consensus problems consensus problems nonlinear receding horizon control optimization introduction sophisticated structure related consensus problems attracted significant interest recent years complex nature sophisticated framework multi agent consensus problem serves fertile email preprint submitted elsevier march ground application advanced control algorithms found basis many areas including cooperative control formation control flocking rendezvous far consensus methodologies widely explored multiagent linear dynamical systems however physical systems nonlinear nature even exhibit phenomenon chaos combination consensus nonlinear dynamics remains major challenge literature although preliminary results presented recent years instance work new rule introduced nonlinear protocol design allowing consensus general set values investigation concentrated nonlinear cooperative control consensus nonlinear heterogeneous systems discussed problem consensus systems inherent nonlinear dynamics directed topologies variable transformation method used convert consensus problem partial stability problem studied consensus nonlinear systems without assumption topology connected fixed hand receding horizon control based methodologies gained significant momentum last two decades used obtain optimal feedback control law minimizing desired performance index given finite horizon sense receding horizon control one powerful methodologies adapted consensus problem dynamics based approach many results developed consensus problems applications work presented distributed receding horizon control law dynamically coupled nonlinear systems based linearization representative robust distributed receding horizon control methods studied nonlinear systems coupled constraints communication delays proposed robust distributed model predictive control methods nonlinear systems subject external disturbances recent study type optimization combined receding horizon control strategies uav formation flight dynamics previously conducted studies nonlinear consensus problem either employ sort adaptation mechanisms adaptive control methodologies simplistic linearization methods paper propose scheme solving nonlinear consensus problem utilizing nonlinear receding horizon control nrhc methodology avoids pitfall utilizes complete architecture nonlinear system given fixed network nonlinear consensus problem converted optimization framework control protocol designed locally agent nonlinear receding horizon control algorithm important extension existing literature presents novelty proposed nrhc procedure optimization algorithm applied avoid high computational complexity large data storage process agent needs obtain neighbor state via given network time instant efficient distributed strategies involve multiple information exchanges predicted trajectories states presents second major novelty proposed approach formulation also provide asymptotic stability guarantees achieve consensus within given topology proposed scheme extended consensus problem nonlinear systems effective nature proposed methodology demonstrated several examples different topologies approach demonstrate applicability distributed nonlinear receding horizon control consensus routine nonlinear systems organization paper following preliminaries problem statement distributed nonlinear receding horizon control presented nonlinear consensus dynamics framework detailed control design investigated stability analysis discussed theoretical aspects proposed methodology applied lorenz oscillators oscillators examples paper concluded preliminaries problem statement section introduce notation used throughout paper provide broad definition main problem studied later present algorithms solving problem work denotes real space real matrix transpose inverse denoted respectively symbol represents kronecker product matrices euclidean norm denoted kxk norm denoted kxkp stands identity matrix dimension given matrix represents matrix positive definite negative definite define column operation col xtn column vectors consider system nonlinear agents agent dynamic system given xin state vector ith oscillator collection agent neighbor states function corresponding nonlinear vector field control input agent function satisfies global lipschitz condition therefore exists positive constant kxi condition satisfied jacobians uniformly bounded exists communication network among agents network described undirected directed graph denotes node set denotes edge set aij adjacency matrix framework exists connection nodes agents aij otherwise aij assume graph aii path sequence connected edges graph path two nodes graph said connected symmetric matrix called undirected graph set neighbors node denoted agent denoted degi aij degree matrix denoted diag degm laplacian matrix described nonlinear system given network topology distributed control protocol said achieve consensus lim kxi collection agent neighbor states mind specific work interested designing distributed control strategy using nonlinear receding horizon control methodology agent achieve consensus within given network topology distributed nonlinear receding horizon control protocol presented framework agent following optimization problem utilized generate consensus protocol locally within network argmin subject performance index designed follows aij kxi aij kxi kui qin symmetric matrices horizon denote describe terminal cost agent utilize control scheme able deal nonlinear nature graph scrutiny desired solve nonlinear optimization problem directly construction cost function given consensus problem converted optimization procedure purpose utilize powerful nature nonlinear receding horizon control algorithm generate distributed consensus protocol minimizing associated cost function context agent needs obtain neighbors information via given network efficient centralized control strategy distributed strategies involve multiple information exchanges predicted trajectories states performance index evaluates performance present time finite future minimized time segment starting structure possible convert present receding horizon control problem family finite horizon optimal control problems artificial axis parameterized time according necessary conditions optimality local problem tpbvp formed follows hui denotes costate agent hamiltonian defined aij kxi kui aij fxi qin aij denotes variable optimal control problem distinguish correspondence original problem notation hxi denotes partial derivative respect methodology since state costate determined tpbvp state costate tpbvp regarded nonlinear algebraic equation respect costate denotes costate actual local control input agent given arg hui formulation optimal control calculated based ordinary differential equation solved numerically without need iterative optimization routine since nonlinear equation satisfied time dpi holds along trajectory system receding horizon control smooth function time solution tracked respect time however numerical errors associated solution may accumulate integration proceeds practice therefore correction techniques required correct errors solution address problem stabilized continuation method used according method possible rewrite statement dpi denotes stable matrix make solution converge zero exponentially evaluate optimal control computing derivative real time consider partial differentation respect time fxi fui hxi hxi hui futi hui since hui futi fxi fui hui fui hxi hxi hui fxti hxi leads following form linear differential equation fxi fui hui fui hxi hxi hui matrix hui nonsingular order reduce computational cost without resorting approximation technique method implemented derivative function respect time given dpi given relationship costate variables assumed follows hxti according following differential equations ati based differential equation costate integrated real time obtained time tpbvp equations integrated forward along axis integrated backward terminal conditions provided next differential equation integrated one step along axis update local optimal controller agent matrix hui nonsingular algorithm executable regardless controllability stabilizability system cost function defined strictly convex guarantees global minimum proof since weighting functions maintain positive definite nature structure qin kkt conditions proposed method gurantees global minima convergence stability analysis sake clarity without loss generality define consensus error col optimal control denoted col cost function written col col order ensure stability proposed nonlinear receding horizon control scheme first consider case terminal cost introduce following definitions regard assume sublevel sets compact path connected moreover use reflect fact cost function quadratically bounded therefore sublevel set dini let sequence upper functions countably compact space suppose sequence decreases monotonically zero convergence uniform let given suppose terminal cost equal zero sampling time exists horizon window receding horizon scheme asymptotically stabilizing proof principle optimality rut sampling time rut since terminal cost equal zero clear implies holds satisfied show example exists yields stability sublevel set contained assured end define sup jts upper semicontinuous clear monotonically decreasing family upper semicontinuous functions defined compact set thus dini theorem stated exists satisfied leading next present stability proposed nonlinear receding horizon control scheme locally quadratic terminal cost let given suppose terminal cost nonnegative locally quadratically bounded sampling time exists horizon window receding horizon scheme asymptotically stabilizing denote cost function zero terminal cost proof use denote cost function locally quadratic terminal cost clear show max also converge uniformly respect locally quadratic positive definite terminal cost theorem uses results consider nonlinear system given assume graph defining topology multi agent system connected given distributed control protocol based exists large enough value horizon guarantees consensus error remain asymptotically stable achieve consensus system although sweep algorithm executable whenever system stable result optimization horizon chosen sufficiently long monotonicity cost function becomes sufficient condition stability therefore also ensure stability nonlinear system distributed nonlinear receding horizon control method extension case section extend proposed scheme consensus problem assume system consisted nonlinear agents leader dynamics agent given leader indexed following dynamical system structure state vector describe connection leader followers leader adjacency matrix diag follower connected leader otherwise new augmented matrix defined nonlinear system given network topology distributed control protocol said achieve consensus follower agents converge leader lim kxi also collection agent neighbor states according presented framework next interested designing distributed control strategy using abovementioned nonlinear receding horizon control methodology agent achieve consensus within given network topology similarly following agent following optimization problem utilized generate consensus protocol locally within network argminji subject corresponding performance index designed follows kxi cij kxi cij kxi kui terminal cost kxi cij kxi qin symmetric trices also horizon utilize framework algorithm solve nonlinear consensus problem directly addition optimization horizon chosen sufficiently long monotonicity cost function becomes sufficient condition stability numerical example simulation results section demonstrate validity feasibility proposed scheme several nonlinear chaotic systems first consider system agents shown fig agent modeled lorenz chaotic system state agent adjacency matrix given initial states given seen fig directed graph lorenz network connected weighting matrices cost function designed qin diag agents stable matrix designed horizon performance index given simulation implemented matlab time step axis time step artificial axis fig depicts trajectories lorenz systems initial conditions defined possible observe proposed distributed realtime nonlinear receding horizon control strategy results consensus clearly demonstrating effectiveness algorithm horizon length kept sufficiently long ensure stability next consider system agents shown fig agent modeled chaotic system represents state agent case adjacency matrix given seen fig undirected graph network connected weighting matrices cost function designed qin diag stable matrix designed horizon performance index given simulation implemented matlab time step axis time step artificial axis fig depicts trajectories systems initial conditions defined corresponding control strategies consensus achieved suggested distributed nonlinear receding horizon control method consider system one leader following agents agent leader also modeled lorenz chaotic system shown augmented adjacency matrix given diagonal elements show connection leader followers weighting matrices cost function designed diag qin diag agents stable matrix designed initial states given horizon performance index given simulation implemented matlab time step axis time step artificial axis fig depicts trajectories lorenz system initial condition corresponding control strategies consensus reached using distributed nonlinear receding horizon control method consider system one leader agent agents agent modeled chen chaotic system state agent adjacency matrix given weighting matrices cost function designed diag qin diag agents stable matrix designed horizon performance index given fig depicts trajectories chen system initial condition corresponding control strategies consensus reached using distributed nonlinear receding horizon control method seen easily example proposed nonlinear receding horizon control methodology working remarkable within given network topology graph agents systems able reach consensus horizon length kept sufficiently long ensure stability conclusions future works paper investigated consensus problem nonlinear systems using distributed nonlinear receding horizon control methodology different previous works solved nonlinear optimal consensus problem directly without need utilization linearization techniques iterative procedures based stabilized continuation method backward sweep algorithm implemented minimize consensus error among agents local control strategy integrated real time provided stability guarantees systems horizon length kept sufficiently long several benchmark examples different topologies demonstrates applicability significant outcomes proposed scheme nonlinear chaotic systems future studies authors intention extend proposed method systems switching topologies embedded communication references murray consensus problems networks agents switching topology ieee trans automat contr dunbar murray distributed receding horizon control multivehicle formation stabilization automatica dunbar distributed receding horizon control dynamically coupled nonlinear systems ieee trans automat contr hui haddad distributed nonlinear control algorithms network consensus automatica keviczky borrelli fregene godbole balas decentralized receding horizon control coordination autonomous vehicle formations ieee trans control systems technology xie network topology communication data rate consensusability systems ieee trans automat contr chen cao consensus directed networks agents nonlinear dynamics ieee trans automat contr andreasson dimarogonas johansson undamped nonlinear consensus using integral lyapunov function proc american control conference acc xie liu necessary sufficient condition group consensus systems appl math comput wang chen adaptive consensus systems unknown nonlinear dynamics entropy wang wang szolnoki perc evolutionary games multilayer networks colloquium phys condensed matter zhao wang wang finding another multiplex networks applied math comp zhao jia consensus stochastic multiagent systems nonlinear dynamics appl math comput duan chen huang consensus multiagent systems synchronization complex networks unified viewpoint ieee trans circuit sys zhu yuan consensus systems switching topologies linear algebra applications movric lewis cooperative optimal control multiagent systems directed graph topologies ieee trans automat contr bauso giarre pisanti protocols optimal distributed consensus networks dynamic agents systems control letters wang nonlinear cooperative control consensus nonlinear heterogeneous systems proc ieee conf decision control liu xie ren wang consensus systems inherent nonlinear dynamics directed topologies systems control letters cao consensus systems jointly connected topology iet control theory appl dunbar distributed receding horizon control algorithm dynamically coupled nonlinear systems proc ieee conf decision control european control conference shi distributed model predictive control constrained nonlinear systems communication delays systems control letters shi robust distributed model predictive control constrained nonlinear systems robustness constraint approach ieee trans automat contr qiu duan receding horizon control multiple uav formation flight based modified brain storm optimization nonlinear dyn bryson applied optimal control hemisphere new york secs kabamas longman homotopy approach feedback stablilization linear systems guidance control dyn ohtsuka fujii stabilized continuation method solving optimal control problems guidance control dyn ohtsuka fujii optimization algorithm nonlinear control automatica ohtsuka control nonlinear systems guidance control dyn antoniou practical optimization algorithm engineering applications springer sec jadbabaie hauser stability receding horizon control general terminal cost ieee trans autom contr lorenz deterministic nonperiodic flow atmos sci chen new chaotic attractor coined int bifurcation chaos chen ueta yet another chaotic attractor int bifurcation chaos figure communication topology lorenz chaotic systems agents sec sec sec sec figure trajectories agents lorenz chaotic system control protocol generated distributed nrhc figure communication topology chaotic systems agents sec sec sec sec figure trajectories agents chaotic system control protocol generated distributed nrhc sec sec sec sec figure trajectories leader following agents lorenz chaotic system control protocol generated distributed nrhc sec sec sec sec figure trajectories leader following agents chen chaotic system control protocol generated distributed nrhc
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log algorithm problem mar haitao jingru department computer science utah state university logan usa department computer science marshall university huntington usa abstract consider classical problem trees let tree vertices every vertex nonnegative weight problem find centers edges maximum weighted distance vertices closest centers minimized megiddo tamir siam gave algorithm solve problem time using cole parametric search since open three decades whether problem solved log time paper present log time algorithm problem thus settle open problem affirmatively introduction paper study classical problem trees let tree vertices edge connecting two vertices positive length consider edge line segment length talk points edge two points unique path denoted slightly abusing notation use denote length vertex associated weight problem compute set points called centers maximum weighted distance vertices closest centers minimized formally value minimized vertex set note center interior edge kariv hakimi first gave log time algorithm problem jeger kariv proposed log time algorithm megiddo tamir solved problem log log time running time algorithm reduced applying cole parametric search progress made recently banik small values log log algorithm another log algorithm given since megiddo tamir work open whether problem solved log time paper settle long open problem affirmatively presenting log algorithm note previous algorithm first algorithm rely cole parametric search involves large constant time complexity due aks sorting network algorithm however avoids cole parametric search center required located vertex call discrete case previously algorithm case runs time techniques also solve discrete case log time preliminary version paper appear proceedings international symposium computational geometry socg related work many variations problem studied problem solvable time path problem already solved log time bhattacharya shi also gave algorithm whose running time linear exponential unweighted case vertices weight log algorithm given problem later megiddo solved problem time algorithm improved log time finally frederickson solved problem time four papers also solve discrete case following problem version running times points considered demand points centers required vertices points demand points centers points megiddo tamir solved problem time running time reduced applying cole parametric search related problems frederickson presented algorithms following tree partitioning problems remove edges maximum minimum total weight connected subtrees minimized maximized finding centers general graph geometric version problem plane also finding centers demanding points plane special cases however solvable polynomial time example problem solved time solved log time also refer faster randomized algorithm require centers given line problem finding centers solved polynomial time recently problems uncertain data studied extensively problem variations uncertain data also considered approach discuss approach problem discrete case similar even simpler let optimal objective value optimal solution feasibility test determine whether given value yes call feasible value given feasibility test done time algorithm follows algorithmic scheme unweighted case similar tree partition problems however big difference three schemes proposed gradually solve problems time approach follows first scheme significantly simplifies algorithm one reason first scheme sufficient algorithm runs log time logarithmic factor feasibility test algorithm contrast efforts last two schemes reduce running time algorithms corresponding feasibility test algorithms specifically algorithm consists two phases first phase gather information feasibility test done faster time using faster feasibility test algorithm second phase computes optimal objective value also use tree addition matrix searching algorithm utilize techniques sublist queries line arrangements searching etc remark might tempting see whether techniques adapted solving problem time unfortunately found several obstacles prevent example key ingredient techniques build sorted matrices implicitly time matrix element obtained time problem however since vertices tree weights elusive achieve goal indeed also one main difficulty solve problem even log time thus makes problem open long time seen later log time algorithm circumvents difficulty combining several techniques based study although proof suspect log lower bound problem rest paper organized follows section review previous techniques used later section describe techniques dealing stem finally solve problem section slightly modifying techniques solve discrete case section preliminaries section review techniques used later algorithm feasibility test given value feasibility test determine whether feasible whether say vertex covered center note feasible place centers vertices covered following describe feasibility test algorithm essentially one although description much simpler pick vertex root denoted vertex use denote subtree rooted following traversal place centers greedy manner vertex maintain two values sup dem sup distance closest center placed dem maximum distance place center within distance uncovered vertices covered also maintain variable count record number centers placed far refer algorithm pseudocode following postinitially count vertex sup dem order traversal suppose vertex visited child update sup dem follows sup dem use center closest cover uncovered vertices thus reset sup min sup sup note since connects edge length edge otherwise dem place center edge distance dem update count sup min sup dem otherwise dem update dem min dem dem root visited sup dem place center update count count finally feasible count algorithm runs time use refer algorithm remark algorithm actually partitions disjoint connected subtrees vertices subtree covered center located subtree make use observation later solve problem key compute find centers applying algorithm feasibility test algorithm input tree root value output determine whether feasible count vertex sup dem vertex traversal child sup dem sup min sup sup else dem count place center edge distance dem sup min sup dem else dem min dem dem sup dem count return true count place center root matrix searching algorithm review algorithm msearch proposed widely used matrix sorted elements every row every column nonincreasing order given set sorted matrices searching range feasible stopping count msearch produce sequence values one time feasibility tests test elements matrices discarded suppose value produced need test feasible updated otherwise updated msearch stop number remaining elements matrices lemma proved slightly change statement accommodate need lemma let set sorted matrices dimension let number feasibility tests needed msearch discard elements max log maxj log total time msearch exclusive feasibility tests log time evaluating matrix element number matrix elements need evaluated log sublist queries let set upper plane given two indices sublist query asks lowest point common intersection version query given vertical line two indices query asks lowest point common intersection lemma proved lemma discussion query algorithm version used procedure proof lemma lemma build data structure log time sublist query answered time query answered log time remark log preprocessing time poly log algorithms query problems would sufficient purpose poly polynomial function line arrangement searching let set lines plane denote arrangement lines let denote vertex let lowest vertex whose feasible value let highest vertex whose smaller definitions vertex lemma proved lemma vertices computed log time time feasibility test remark alternatively use cole parametric search compute two vertices first sort lines intersections horizontal line done log time cole parametric search found additional time intersection two adjacent lines sorted order line arrangement searching technique modified slope selection algorithms avoids cole parametric search following often talk problems plane context clear point use denote respectively algorithms stems section first define stems similar spirit proposed unweighted case present two algorithms solving problem stem techniques used later solving problem tree let path vertices denoted sorted left right vertex incident edges two additional edges connecting two vertices either vertex may exist let denote union additional edges see fig two points still use denote unique path use denote length path fig illustrating stem respect range call stem following holds exists exists following terminology call thorn twig called thorn vertex called twig vertex called backbone vertex called backbone vertex define length total number vertices remark algorithm section produce stems defined path vertices also vertices however thorn may original edge corresponds path sense length equal distance also case twig algorithm section maintain range feasible feasible since feasibility test made value definitions thorns twigs imply following thorn vertex place center backbone cover twig vertex need place center edge cover sequel give two different techniques solving problem stem fact algorithm problem section use techniques process stem rather directly solve problem let temporarily refer optimal objective value problem rest section assume first algorithm algorithm motivated following easy observation exist two vertices center located path since otherwise could move achieve situation assume backbone vertices system origin defines two lines containing slopes respectively see fig thorn also defines two lines follows define uli uri point hence uli uri left right distance define line uli slope line uri slope note intersect point whose whose equal twig vertex define points wil wir lines way uli uri fig illustrating definitions lines defined backbone vertex thorn vertex consider point backbone right side verified weighted distance exactly equal intersection vertical line left side similar observation also true let denote set lines defined vertices note based observation equal vertex line arrangement precisely equal vertex defined section lemma compute log time second algorithm algorithm relies algorithm msearch first form set sorted matrices define two lines section exists also define otherwise refer let denote respectively four upper bounded four lines index order arbitrary way set upper define lowest point common intersection upper upper halfplanes defined observe use one center cover backbone thorn vertices equal optimal objective value problem define matrix dimension otherwise twig define two arrays ari ali elements follows let denote respectively upper bounded lines defined section array ari defined vertices right side follows use single center cover vertices ari defined optimal objective value problem equal lowest point common intersection upper symmetrically array ali defined left side specifically use one center cover vertices defined optimal objective value equal lowest point common intersection upper let set matrices ari ali following lemma implies apply msearch compute lemma matrix sorted element matrix proof first show matrices sorted consider matrix consider two elements row goal show zero thus trivially holds way defining upper one verify therefore let set upper since subset thus lowest point common intersection upper higher hence thus proves therefore elements row sorted nonincreasing order similar approach show elements column also sorted nonincreasing order omit details hence sorted matrix consider array ali consider two elements ali ali goal show ali ali argument similar third case let set upper since subset lowest point common intersection upper higher hence ali ali show ari also sorted similar way omit details proves every matrix sorted following show must element one matrices imagine apply feasibility test algorithm stem considering tree root algorithm compute centers algorithm actually partitions disjoint connected subtrees vertices subtree covered center located subtree must subtree center two vertices since otherwise could adjust positions centers maximum weighted distance vertices closest centers would strictly smaller since connected path also depending whether one twig vertex two cases neither vertex twig vertex claim thorn vertices connecting backbone vertices covered center indeed suppose backbone vertex connects thorn vertex assume contrary covered recall definition thorns since according covered center hence connected subtree denoted partition induced clearly since connected every vertex must thus however since also incurs contradiction since proves claim backbone vertex let index otherwise thorn vertex let index connects backbone vertex similarly define without loss generality assume claim implies equal lowest point common intersection upper defined backbone vertices thorn vertices thus equal therefore matrix next consider case least one twig vertex twig vertex definition since twig must contain center covered one twig vertex since otherwise would need two centers cover since twig must contain center without loss generality assume twig vertex say backbone vertex let index otherwise thorn vertex let index connects backbone vertex without loss generality assume argument thorn vertices covered implies lowest point common intersection upper defined backbone vertices thorn vertices thus ari therefore array ari proves must matrix lemma thus follows note consists matrix dimension arrays lengths help sublist query data structure lemma following lemma shows matrices implicitly formed log time lemma log time preprocessing matrix element evaluated time proof build sublist query data structure lemma upper log time element computed time sublist query consider array ali given index compute ali recall ali equal lowest point common intersection upper set upper lowest point common intersection upper computed time sublist query query indices computing also done time slightly modifying query algorithm computing briefly discuss interested reader refer details proof lemma discussion lemma query algorithm computing similar spirit algorithm linear programming problem binary search algorithm iteration algorithm computes highest intersection vertical line bounding lines halfplanes based local information intersection algorithm determine side proceed search computing need incorporate additional end iteration binary search compute highest intersection compare intersection bounding line update fig left tree numbers weights vertices right path partition highest intersection needed costs constant extra time iteration therefore total running time computing still computing elements arrays ari done similarly lemma thus follows applying algorithm msearch stopping count according lemma msearch produces log values feasibility tests total time exclusive feasibility tests need evaluate log matrix elements hence total time computing log remark clearly first algorithm better second one however later use techniques second algorithm often bounded thus log fact use techniques second algorithm mainly need set stopping count value solving problem section present algorithm solving problem focus computing optimal objective value frederickson proposed partition edges paths vertex endpoint path degree equal see fig path called contains leaf generalize follows course algorithm range contains maintained modified removing edges adding thorns twigs point algorithm let thorns twigs removed stem path along thorns twigs connect vertices path partition stems according stem called contains leaf backbone vertex stem algorithm follows first algorithmic scheme two main phases phase phase let phase gathers information feasibility test made sublinear time phase computes using faster feasibility test leaves additional phase called phase reduces problem tree leaves phase part phase separates phase make clearer following consider general case leaves algorithm gives pseudocode overall algorithm preprocessing computing vertex ranks first perform preprocessing recall root compute distances vertices time ancestor computed time following whenever need compute distance always case one ancestor thus obtained time next compute rank rank vertex ranks facilitate algorithm later vertex define point equal system define line slope equal let set lines consider line arrangement let vertices defined section lemma vertices computed log time let horizontal line strictly sort lines intersections left right vertex define rank lines order definitions order also order sorted intersections horizontal line phase recall leaves section reduce problem problem placing centers tree leaves algorithm maintain range contains initially already computed preprocessing form actually since thorns twigs initially done time recall leaves following recall length stem defined number backbone vertices let set whose lengths let number backbone vertices form matrices lemma let denote collection matrices call msearch stopping count using feasibility test algorithm msearch stops updated range matrix elements called active values since active values remain thus active values without active values perform following procedure backbone vertex closest root called top vertex place centers subtract number replace either thorn twig connected top vertex thus removed except top vertex solving problem modified also solves problem original procedure implemented time length details given procedure let top vertex run feasibility test algorithm root arbitrary value finally processed depending whether sup dem following sup dem let last center placed case vertices covered covered according algorithm discussed proof lemma covers connected subtree vertices let denote set vertices excluding note easily identified let number centers excluding placed since matrices formed based active values following key observation run algorithm also cover vertices centers cover vertices one center indeed true way form matrices consistent discussed proof lemma case replace attaching twig length equal vertex following property place center path distance cover vertices dominates vertices thus sufficient keep since feasible subsequent feasibility test algorithm use following lemma shows vertex largest rank lemma let vertex largest rank following holds covers vertices point path distance proof prove let point distance add dummy edge extended root long enough also proves later show first show covers vertices consider vertex goal prove trivially holds following assume note may twig twig means holds case run center placed cover hand according key observation use one center cover vertices hence covers vertices thus covers following assume twig note definition thorns thorn thus must either backbone outside define set vertices subtree rooted let depending whether two cases case recall section vertex defines line consider two lines let denote intersections horizontal line respectively see fig note point corresponding point sense since rank rank definition ranks holds slope positive intersection vertical line hand since exactly equal therefore obtain case case definition must according key observation use one center cover vertices particular use one center cover definition closest point cover hence must able cover therefore obtain fig illustrating proof case note points defined respectively section proves finally argue assume contrary true outside means place center outside cover vertices contradicts key observation place center cover vertices lemma thus follows due preprocessing section find time finishes procedure case sup dem since thus indeed twig next consider case sup dem case vertices covered yet would need place center cover let set uncovered vertices identified case replace attaching thorn length equal vertex following property center outside covering mean contains distance also covers vertices intuitively dominates vertices since later place centers outside cover vertices sufficient maintain following lemma shows vertex largest rank lemma let vertex largest rank center outside covers distance also covers vertices proof let vertex goal prove proof similar case lemma omit details since center would cover holds implies thus indeed thorn replaces attaching either thorn twig perform following additional processing suppose attached thorn already another thorn discard one whose rank smaller center covers remaining vertex cover discarded one well proof similar lemma omit makes sure one thorn suppose attached twig already another twig discard one whose rank larger subtract reason following without loss generality assume rank rank since twigs fig illustrating proof lemma note points defined respectively section apply algorithm place center distance place center distance rank rank following lemma lemma proof analysis similar lemma consider lines defined respectively discussed section let intersections horizontal line respectively see fig since rank rank note since hand due thus obtain lemma thus follows lemma tells vertex covered subsequent algorithm also covered thus sufficient maintain twig since need place center subtract removing hence one twig finishes procedure due preprocessing section running time procedure let modified tree stem without active values still leaves repeat algorithm stops leaves finishes phase following lemma gives time analysis excluding preprocessing section lemma phase runs log log time proof first argue number iterations loop log analysis similar include completeness consider iteration loop suppose number denoted least length larger hence least half length thus recall total number backbone vertices active values msearch least removed note removing two may make interior vertex become new leaf modified tree hence tree resulting end iteration tree beginning iteration therefore number iterations loop needed reduce number log proceed analyze running time phase iteration loop call msearch matrices since length stem matrices formed perform preprocessing lemma matrices matrix element evaluated time total time preprocessing stems log since matrices stopping account call msearch produces log values feasibility tests time log matrix elements evaluated without active values time hence total time iteration since log iterations total number feasibility tests thus overall time feasibility tests phase hand iteration active values deleted since length active values backbone vertices thus least backbone vertices deleted iteration therefore total sum iterations hence total time preprocessing lemma log total time msearch total time without active values summary overall time phase excluding preprocessing section log log since phase assume tree leaves want place centers cover vertices note may thorns twigs main purpose phase gather information feasibility test done sublinear time specifically time recall range contains first form partition stems substems length lowest backbone vertex substem highest backbone vertex next lower substem thorn twig included upper substem results partition edges let set substems let tree node represents substem node parent node highest backbone vertex substem lowest backbone vertex substem call stem tree since leaves number nodes substem compute set lines section let set lines substems define lines system clearly consider line arrangement define vertices section lemma vertices computed log time update max min hence still call values active values substem observe element matrices formed based section equal intersection two lines thus equal vertex definitions matrix element active wil fig illustrating proof lemma note wil point defined discussed section future algorithm need test feasibilities values follows compute data structure substem help make feasibility test faster prove following lemma use denote feasibility test algorithm lemma lemma log time preprocessing feasibility test done log time first discuss preprocessing present algorithm preprocessing consider substem let backbone vertices sorted left right top vertex vertex may twig thorn let arbitrary value following statements made applicable due none elements matrices produced active definition twigs run algorithm place center denoted twig distance first run following cleanup procedure remove vertices covered centers twigs cleanup procedure first compute rank rank line follows let sequence lines sorted intersections horizontal line left right definitions also sequence lines sorted intersections horizontal line fact sequence unique line lines define clearly rank lines computed time consider twig backbone vertex recall defines line slope defines line slope thus following lemma lemma center covers rank proof let intersections horizontal line respectively refer fig let denote intersection vertical line since located according definitions exactly equal hence covers line hand line right rank lemma thus follows consider thorn vertex recall defines line slope similarly covers rank based observations use following algorithm find backbone thorn vertices covered centers twigs left sides let smallest index twig algorithm maintains index initially incrementally following reset rank reason rank rank covered center twig also covered center twig thus sufficient maintain twig next rank mark covered exits mark covered algorithm runs time marks vertices exists twig whose center covers symmetric way scanning vertices mark time vertices exits twig whose center covers omit details marks vertices covered centers twigs let set backbone thorn vertices marked vertices need covered placing centers backbone outside thorn vertex connected backbone vertex means covered center twig covered center observe center backbone outside covers cover well convenience discussion include well let backbone vertices sorted left right closer root note may may thorn vertex use denote finishes cleanup procedure next compute data structure maintain information faster feasibility tests first maintain index twig vertex rank rank twig vertex reason keep following observe vertex descent vertex another substem descent substem stem tree covered center twig also covered center twig symmetrically maintain index twig vertex twig vertex similarly vertex substem subtree rooted covered center twig also covered center twig computed time two indices use denote set backbone vertices thorn vertices index maintain integer ncen vertex define roughly speaking ncen minimum number centers needed cover vertices minus one use ncen centers cover many vertices possible left right vertex covered dominates uncovered vertices detailed definitions given let smallest index possible cover vertices one center index exist let define ncen define vertex rank vertex reason maintain follows suppose feasibility test vertices covered need place new center cover according greedy strategy want place center close root possible two cases first case one verify place center top vertex cover vertices case place center backbone use center outside cover precisely center outside subtree rooted substem maintain center outside covering cover vertices well second case need place center backbone according greedy strategy want place center close much possible use denote center following lemma shows determined lemma path distance proof proof somewhat similar lemma note since let point distance point must exist definition point closest cover backbone vertex backbone otherwise thorn vertex backbone vertex connects since obtain thus must backbone hence either case backbone consider vertex following show covered subtree rooted contains since one verify covered otherwise assume contrary cover would need move towards order cover however since point closest cover also point closest cover hence moving towards make cover implies point backbone cover contradicts fact possible place center backbone cover vertices therefore covers shows point backbone closest cover vertices thus lemma follows defines ncen case define ncen recursively ncen ncen note thus recursive definition valid following present algorithm compute ncen fact recursive definition implies dynamic programming approach scan vertices backward details given following lemma lemma ncen computed time proof consider following problem find center cover backbone thorn vertices discussed section backbone thorn vertex defines two upper optimal objective value problem equal lowest point common intersection defined backbone thorn vertices section preprocessing first compute upper defined vertices order indices corresponding vertices compute sublist query data structure lemma log time section lowest point common intersection upper defined vertices computed sublist query time use denote optimal objective value problem preprocessing given computed time proceed compute ncen downto following maintain index initially set ncen process index follows first compute time depending whether two cases depending whether two subcases ncen ncen otherwise first set ncen rank rank reset thorn reset keep decrementing one reset ncen ncen difficult see algorithm runs time time since since compute data structure substem log log time total time computing data structure substems log log data structures show feasibility test done time faster feasibility test given goal determine whether feasible work stem tree node represents stem initially set sup dem sup dem infinitely large value still smaller sup every stem perform traversal maintain variable count number centers placed far suppose processing stem child stem reset sup min sup sup dem min dem dem handling children process follows first increase count number twigs let uncovered vertices defined backbone vertices note recall maintained two twig indices depending whether sup dem two main cases case sup dem sup dem uncovered vertices children covered center determines value sup child stem sup lowest backbone vertex note need compute use discussion binary search list find largest index cover vertices exists let index found time using sublist queries lemma shown following lemma lemma index largest index cover vertices found time proof given index show determine whether cover vertices log time recall preprocessing vertex defines two upper built sublist query data structure upper defined vertices let point equal let vertical line let lowest point common intersection upper defined vertices observation cover vertices determined log time sublist query cover vertices continue search indices larger otherwise continue search indices smaller cover vertices return total time vertices covered case reset sup min sup latter value distance center twig includes case increase one reset count reset dem sup otherwise need place additional center cover uncovered vertices including thus increase count one reset sup min dem case sup dem case need first deal dem covering vertices children stems covered dem center twig cover uncovered vertices children stems case increase count ncen postpone placing centers next stem reset dem sup otherwise increase count one reset sup min dem dem binary search find largest index find center backbone cover vertices dem index exit let following lemma shows index found time lemma index found time proof given index first show determine time answer following question whether exists center backbone cover vertices dem sublist query upper defined vertices compute lowest point common intersection answer question otherwise dem answer question yes dem let vertical line whose dem sublist query compute lowest point common intersection upper log time answer question yes otherwise answer total time determine answer question answer yes continue search indices larger otherwise continue indices smaller answer question return total running time two subcases dem postpone placing centers next stem resetting dem min dem sup otherwise place center backbone distance max dem increase count one reset sup min dem includes case place center location determined algorithm lemma cover dem well vertices increase count one next increment one increase count ncen reset dem sup otherwise increase count one reset sup min dem finishes processing stem stem contains root processed sup dem place center root cover uncovered vertices increase count one value feasible count since spend time stem stems runs time refer algorithm pseudocode phase phase finally compute optimal objective value using faster feasibility test recall computed range contains phase first form one following let set stem compute set lines section let set lines stems lemma compute two vertices arrangement defined section update max min discussed phase stem active values matrices defined next stem perform procedure section place centers subtract number replace attaching twig thorn top vertex let modified tree loop single stem apply algorithm stem obtained value running time phase bounded log analyzed following theorem theorem problem solved log time algorithm faster feasibility test algorithm input original input output determine whether feasible count stem stem tree sup dem sup stem traversal child sup min sup sup dem min dem dem increase count number twigs let set uncovered vertices backbone vertices sup dem let center child stem gives value sup binary search find largest index cover vertices sup min sup else count count ncen dem sup else count sup min dem else dem count count ncen dem sup else count sup min dem else binary search find largest exists center backbone cover vertices dem dem dem min dem sup else count max dem sup min dem else count count ncen dem sup else count sup min dem sup dem count return true count proof discussed phases run log time focus phase first number iterations loop log number halved iteration iteration let denote total number backbone vertices hence thus call lemma takes log time total time procedure since removed iteration total sum phase therefore total time algorithm lemma phase log log since also overall time procedure phase therefore total time phase log proves theorem pseudocode algorithm summarizes overall algorithm algorithm algorithm input tree integer output optimal objective values centers perform preprocessing section compute ranks vertices phase log form leaves let set lengths form set matrices lemma let total number backbone vertices call msearch stopping count using active values perform place centers subtract number replace thorn twig modify phase stem partition substems lengths let set substems form compute set lines stems way discussed section compute two vertices lemma update substem compute data structure faster feasibility test phase form one compute set lines stems way discussed section compute two vertices lemma update perform place centers subtract number replace thorn twig modify compute set lines compute two vertices lemma update apply find centers original tree discrete problem section extend techniques solve log time discrete problem centers must located vertices fact problem becomes easier due following observation observation optimal objective value equal two vertices center placed cover previous time algorithm relies observation megiddo first computed time collection log sorted subsets contain intervertex distances pairs vertices multiplying weight elements subsets corresponding vertex contained new sorted subsets computed time searching sorted subsets using msearch frederickson johnson later proposed log algorithm computes succinct representation intervertex distances using sorted cartesian matrices msearch algorithm solves unweighted case problem log time however techniques may generalized solving weighted case multiply vertex weights elements cartesian matrices new matrices sorted guarantee columns rows sorted different rows columns multiplied weights different vertices algorithm uses similar techniques previous problem following briefly discuss mainly focus pointing differences first need modify feasibility test algorithm difference sup dem dem instead placing center interior edge place center update sup min sup use replace line pseudocode algorithm running time still use denote new algorithm algorithm stems stem defined slightly differently suppose range contains backbone vertex stem still one thorn one twig thorn defined way however twig consists two edges means place center cover still call twig vertex following terminology call bud next give algorithm solve problem stem backbone vertices algorithm similar section uses msearch use different way form matrices based observation note need similar algorithm section let temporarily refer optimal objective value problem subsection assume let backbone vertices assume backbone vertices origin section thorn vertex define two points uli uri whose weights equal bud twig vertex let set vertices hence sort vertices left right let sorted list use denote vertex define two sorted arrays ali ari lengths follows define ari define ali arrays sorted let denote set sorted arrays defined observation must element array log time preprocessing array element computed time applying msearch stopping count using compute log time msearch produces log values feasibility tests log time solving problem sequel solve discrete problem first preprocessing section three phases let phase assume leaves since otherwise could skip phase phase except following changes first use replace second form matrix set way discussed section third without active values modify procedure follows let top vertex run root value processed depending whether sup dem two main cases case sup dem sup dem define way include bud vertices difference vertex note case need place center let vertex makes center specifically refer pseudocode algorithm place center vertex line let vertex determines value dem dem note must descendent case replace twig consisting two edges lengths equal respectively find vertex one way modify vertex determines value dem vertex also maintained another way fact vertex largest rank proved following lemma whose proof similar lemma note actually consistent way creating twigs previous case lemma vertex largest rank proof let vertex goal show rank rank note either backbone vertex bud first discuss case backbone vertex define set vertices subtree rooted let depending whether two subcases assume contrary rank rank recall defines line defines line preprocessing see section refer fig let fig illustrating proof lemma note points defined respectively preprocessing section denote intersections horizontal line respectively since definition ranks note corresponds point sense actually point closest cover similarly corresponds point since one verify contradicts vertex makes center determines value dem descendants first note twig vertex since otherwise would need bud order cover depending whether backbone vertex thorn vertex two subcases fig illustrating proof lemma fig illustrating proof lemma backbone vertex ancestor let intersections horizontal line respectively see fig since determines center ancestor according holds since ancestor also ancestor therefore point intersection must left vertical line since slope positive obtain rank rank thorn vertex assume contrary rank rank let backbone vertex connects since backbone vertex must ancestor rank rank following prove implies cover thus incurs contradiction preprocessing section vertex defines point see fig little abuse notation use denote define way since determines ancestor according holds implies since rank rank thus intersection vertical line larger note equal therefore obtain since path contains thorn thus holds proves rank rank case backbone vertex next discuss case bud since descendent must twig vertex twig let backbone vertex connects say either contains backbone vertex contains backbone vertex connects thorn vertex otherwise analysis similar subcase omit details following analyze case although may either backbone vertex thorn vertex prove rank rank uniform way assume contrary rank rank refer fig define points figure way except backbone vertex connects goal show incurs contradiction since covers end since sufficient show proof similar subcase thorn vertex briefly discuss since determines ancestor holds implies since rank rank thus intersection vertical line larger equal therefore proves lemma replaces attaching twig addition already another twig bud discard new twig discard old twig otherwise guarantees one twig case sup dem sup dem define way excluding buds let vertex largest rank lemma dominates vertices thus replace thorn whose length equal addition already another thorn reason discard one whose rank smaller finishes procedure running time still similar analysis lemma phase still runs log time omit details phase first still form node represents substem length instead using line arrangement searching technique resort msearch let set substems let set matrices substems formed way described section apply msearch stopping count using since arrays lengths msearch produce log values feasibility tests log time total feasibility test time log since msearch stops updated range matrix element active let arbitrary value compute data structure stem feasibility test made sublinear time show log time preprocessing feasibility test done time let substem backbone vertices top vertex preprocessing algorithm works similar way cleanup procedure first perform cleanup procedure remove vertices covered centers buds twigs note buds twig vertices automatically covered centers buds need find backbone thorn vertices covered buds easier locations centers fixed buds use following algorithm find backbone thorn vertices covered buds left sides let smallest index bud algorithm maintains index initially incrementally following bud reset covered also covered next mark covered exits mark covered note although ancestor descendent still compute time latter two distances computed time due preprocessing section algorithm runs time symmetric way scanning right left also mark backbone thorn vertices covered buds right sides finishes cleanup procedure define way let backbone vertices also define way computing data structure sequel compute data structure maintain index twig twig index maintain index twig twig index found time also compute integer ncen vertex whose definitions similar addition maintain another vertex details given define similarly smallest index possible cover vertices center located vertex index exist let define ncen define vertex largest rank point definition consistent based two vertex rank rank rank rank analysis dominates vertices define follows undefined otherwise largest index therefore refers index backbone vertex closest cover definition also cover vertices proof similar lemma omit define ncen ncen compute ncen observe known computed additional log time binary search backbone vertices therefore focus computing ncen use similar algorithm lemma end need solve following subproblem given two indices want compute optimal objective value denoted discrete problem vertices done time using sublist query data structure shown following lemma lemma log time preprocessing compute time two indices proof preprocessing log time build sublist query data structure upper defined vertices way given indices sublist query compute time lowest point common intersection upper defined vertices let point backbone corresponding note essentially optimal center problem vertices since considering discrete case optimal center discrete problem found follows located vertex equal otherwise let two backbone vertices immediately left right sides respectively either boundary upper envelope bounding lines upper defined vertices convex compute following let line whose equal defined similarly let lowest point define similarly equal lower point center also determined correspondingly found binary search backbone vertices log time point found sublist query log time since compute time preceding lemma use similar algorithm lemma compute ncen time computed additional log time binary search recall hence preprocessing time log log time since total time computing data structure substems log log data structures show feasibility test done time faster feasibility test algorithm similar explain differences referring pseudocode given algorithm first still implement line time exactly algorithm lemma binary search line since constraint must backbone vertex need modify algorithm lemma follows show given index determine time whether center backbone vertex cover vertices dem first compute center optimal objective value discrete problem vertices done time shown proof lemma answer otherwise dem answer yes dem let largest index dem found log time binary search backbone vertices let vertical line whose equal sublist query algorithm faster feasibility test algorithm input original input output determine whether feasible count stem stem tree sup dem sup stem traversal child sup min sup sup dem min dem dem increase count number twigs let set uncovered vertices backbone vertices sup dem let center child stem gives value sup binary search find largest index cover vertices sup min sup bud maintained else count count ncen dem sup else count sup min dem else dem bud maintained count count ncen dem sup else count sup min dem else binary search find largest exists center backbone vertex cover vertices dem dem dem min dem sup else binary search find largest dem count max sup min dem else count count ncen dem sup else count sup min dem sup dem count return true count compute lowest point common intersection upper defined vertices answer yes hence time determine answer question therefore time implementing line addition easy see time binary search line log therefore processing takes time total time phase phase similar following changes first use replace second use new procedure third instead using line arrangement searching technique use msearch specifically pseudocode algorithm replace lines also lines following form matrices way discussed section let denote set matrices call msearch stopping count running time three phases still log shown theorem theorem discrete problem solved log time proof analysis similar theorem briefly discuss since phase phase run log time discuss phase number iterations loop log hence log calls msearch call msearch produces log values feasibility tests therefore total number feasibility tests total time feasibility tests log iteration let denote total number backbone vertices according discussion section call msearch takes log time excluding time feasibility tests since matrix element obtained time total sum phase overall time msearch phase log also overall time procedure phase therefore total time phase log proves theorem references agarwal phillips efficient algorithm euclidean outliers proceedings annual european conference algorithms esa pages ajtai log sorting network proc annual acm symposium theory computing stoc pages banik bhattacharya das kameda song problem tree networks revisited proc scandinavian symposium workshops algorithm theory swat pages bhattacharya shi optimal algorithms weighted problems real line small proc international workshop algorithms data structures pages brass knauer shin vigneron aligned problem international journal computational geometry applications chazelle optimal slope selection via cuttings computational geometry theory applications chan planar algorithms computational geometry theory applications chandrasekaran tamir polynomially bounded algorithms locating tree mathematical programming chen wang efficient algorithms problem theoretical computer science chen wang approximating points piecewise linear function algorithmica chen wang note searching line arrangements applications information processing letters cole slowing sorting networks obtain faster sorting algorithms journal acm cormode mcgregor approximation algorithms clustering uncertain data proc symposium principles database systems pods pages frederickson johnson generalized selection ranking sorted matrices siam journal computing frederickson optimal algorithms tree partitioning proc annual symposium discrete algorithms soda pages frederickson parametric search locating supply centers trees proc international workshop algorithms data structures wads pages frederickson johnson finding kth paths generating searching good data structures journal algorithms huang stochastic problems proceedings symposium discrete algorithms soda pages jeger kariv algorithms finding weighted tree relatively small networks kariv hakimi algorithmic approach network location problems siam journal applied mathematics karmakar das nandy bhattacharya variations constrained minimum enclosing circle problem journal combinatorial optimization katz sharir optimal slope selection via expanders information processing letters megiddo algorithms linear programming related problems siam journal computing megiddo supowit complexity common geometric location problems siam journal comuting megiddo tamir new results complexity problems siam computing megiddo tamir zemel chandrasekaran log algorithm longest path tree applications location problems siam computing wang zhang uncertain data theoretical computer science wang zhang problems plane international journal computational geometry applications wang zhang note computing center uncertain data real line operations research letters wang zhang computing center uncertain points tree networks algorithmica wang zhang covering uncertain points tree proc algorithms data structures symposium wads pages
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joint model referring expressions licheng hao tan mohit bansal tamara berg department computer science university north carolina chapel hill apr licheng airsplay mbansal tlberg abstract expressions apply target object objects image goal corresponds gricean maxim manner one tries clear brief orderly possible avoiding obscurity ambiguity avoiding ambiguity important generated expression easily uniquely mapped target object example two pens table one long red short red asking red pen would ambiguous asking long pen would better reinforcer module incorporated using reinforcement learning inspired behavioral psychology says agents operating environment take actions maximize expected cumulative reward case reward takes form discriminative classifier trained reward speaker generating less ambiguous expressions referring expressions natural language constructions used identify particular objects within scene paper propose unified framework tasks referring expression comprehension generation model composed three modules speaker listener reinforcer speaker generates referring expressions listener comprehends referring expressions reinforcer introduces reward function guide sampling discriminative expressions listenerspeaker modules trained jointly learning framework allowing modules aware one another learning also benefiting discriminative reinforcer feedback demonstrate unified framework training achieves results comprehension generation three referring expression datasets project demo page https within realm referring expressions two tasks computationally modeled mimicking listener speaker roles referring expression generation speaker requires algorithm generate referring expression given target object visual scene shown fig referring expression comprehension listener requires algorithm localize image described given referring expression shown fig introduction people often use referring expressions everyday discourse unambiguously identify indicate particular objects within physical environment example one might point person crowd referring man blue shirt might ask someone pass red pen examples pragmatic interaction two people person intelligent agent robot first speaker must generate expression given target object surrounding world context second listener must interpret comprehend expression map object environment therefore paper propose trained framework models behaviors jointly addition listener speaker also introduce new reinforcer module learns discriminative reward model help generate less ambiguous expressions referring expression generation reg task studied since many early works space focused relatively limited datasets using synthesized images objects artificial scenes limited sets objects simplified environments recently research focus shifted complex natural image datasets expanded include referring expression comprehension task well interactions robotics one reason become feasible several reg datasets collected scale deep learning models applied kazemzadeh introduced first reg dataset natural images imageclef dataset using interactive game later expanded figure joint generation examples using full model three datasets sentence shows generated expression one depicted objects color coded indicate correspondence figure example comprehension results using full model three datasets green box shows region blue box shows correct comprehension based detected regions ages mscoco dataset addition mao collected google reg dataset also based mscoco images setting resulting complex lengthy expressions paper focus evaluations three recent datasets collected mscoco images lstm model previously applied referring expression comprehension task recent work also used model speaker embedding model listener referring expression generation abstract images offline listener reranks speaker output paper propose unified model jointly learns speaker listener models generation comprehension tasks additionally add discriminative reinforcer guide sampling discriminative expressions improve final system instead working independently let speaker listener reinforcer interact resulting improved performance generation comprehension tasks results evaluated three standard datasets verify proposed model significantly outperforms comprehension task tables generation task evaluated using human judgements table automatic metrics table recent neural approaches referring expression generation comprehension tasks roughly split two types first type uses encoderdecoder generative model generate decode sentences given encoded target object careful design visual representation target object model generate unambiguous expressions models referring expression target object easily converted via bayes rule used address comprehension task selecting largest posterior probability second type approach uses model projects visual representation target object semantic representation expression common space learns distance metric generation comprehension performed embedding target object expression embedding space retrieving closest expression object space type approach typically achieves better comprehension performance related work recent years witnessed rise multimodal research related vision language given individual success area need models advanced cognition capabilities several tasks emerged evaluation applications including image captioning visual question answering referring expression image captioning aim image captioning generate sentence describing general content image recent approaches use deep learning address problem perhaps common architecture cnnlstm model generates sentence conditioned visual information image one paper related work glstm uses cca semantics guide caption generation step beyond image captioning locate regions described captions visual genome collected captions dense regions image used tasks many approaches achieving quite good results captioning one challenge caption system output image task dependent therefore movement toward focused tasks visual question answering referring expression generation comprehension involve specific within image discussed referring expression datasets reg studied many years linguistics natural language processing mainly focused small artificial datasets kazemzadeh introduced first dataset refclef using natural images dataset collected game first player writes referring expression given indicated target object second player shown image expression click correct object described expression click lies within target object region sides get points roles switch using game interace authors collected refcoco datasets mscoco images refcoco datasets contain referred objects referring expressions average main difference refcoco players forbidden using absolute location words left dog therefore focusing referring expression purely descriptions addition mao also collected referring expression dataset refcocog using mscoco images framework expressions similar mscoco captions longer complex time constraint data collection setting dataset refcocog objects expressions per object average referring expression comprehension generation referring expressions associated two tasks comprehension generation comprehension task quires system select region described given referring expression address problem model looks object maximizing probability people also try modeling directly using embedding model learns minimize distance paired object sentence embedding space generation task asks system compose expression specified object within image previous work used rulebased approaches generate expressions fixed grammar pattern recent work followed cnnlstm structure generate expressions speaker listener typically considerred formulate two tasks speaker model used referring expression generation listener used comprehension golland proposed optimal speaker act based listener response mao followed idea suppressing ambiguity listener modeling related work uses speaker model generate expressions abstract images uses offline listener rerank speaker output compared model jointly trains speaker listener modules additional reinforcer module encourage unambiguous speaker expression generation moreover use models generation comprehension tasks three natural image datasets model model composed three modules speaker sec listener sec reinforcer sec training speaker listener trained jointly benefit reinforcer reward function reinforcer differentiable incorporated training using reinforcement learning policy gradient algorithm speaker speaker module follow previous use framework cnn model used define visual representation target object visual context term memory lstm used generate likely expression given visual representation improved quantitative performance use visual comparison model speaker encode target object visual representation includes target object context features two visual comparison features specifically target object representation modeled features vgg network global context modeled features extracted layer entire image representation target object modeled vector encoding locations reinforcer sampling reward loss lstm generation loss speaker concat listener man middle wearing yellow lstm mlp mlp embedding loss figure framework speaker model generates referring expression target object listener model learned minimize distance paired object expression representations addition reinforcer module helps improve speaker sampling discriminative less ambiguous expressions training model jointly trained loss functions generation loss embedding loss reward loss thereby improving performance comprehension generation tasks top left bottom right corners target object bounding box well bounding box size respect image xwtl yhtl xwbr yhbr referring expressions often relate object objects type within image red ball blue ball larger elephant comparisons tend quite important differentiation comparison features composed two parts appearance similarp ity koii number objects chosen comparisons location size similarity concatenating difference compared object wtli htli wbri hbri wji hji final visual representation target object concatenation features followed layer fusing together joint feature fed lstm referring expression generation training minimize negative max log log first term second term encourages target object better described true expression compared expressions describing objects image ranking loss expressions listener use model mimick listener behaviour purpose embedding model encode visual information target object semantic information referring expression joint embedding space embeds vectors visually semantically related closer together space referring expression comprehension task given referring expression representation listener embeds joint space selects closest object embedding space predicted target object illustrated fig listener log within image ranking loss objects generalize idea incorporate two triplet hinge losses composed positive match two negative matches given positive match sample contrastive pair expression describing object pair object image optimize following loss max log log log rit note speaker modeled using form structure mao proposed add maximum mutual information mmi constraint encouraging generated expression describe target object better objects model outlined red dashline use lstm encode input referring expression visual representation speaker encode target object thus connecting speaker listener add two mlps perceptions two normalization layers following view object expression mlp composed two fully connected layers relu nonlinearities serving transform object view expression view common embedding space two normalized representations computed similarity score space listener force similarity target object referring expression pairs applying hinge loss triplets consist positive match two negative matches max max thing left choose reward function encourages speaker sample less ambiguous expressions illustrated fig outlined dashed orange reinforcer module learns reward function using paired objects expressions use visual representation target object use another lstm encode expression representation rather using two mlps encode view listener concatenate two views feed together mlp learn logistic regression score trained loss reward function computes match score input object expression use score reward signal eqn sampled expression target object pairs training reward function fixed assist joint system joint model subsection describe specifics three modules speaker listener reinforcer integrated joint framework shown fig listener notice visual vector embedding space learned capture neighbourhood vectors referring expressions thus making aware listener knowledge therefore take mlp embedded vector additional input speaker encodes listener based information fig use concatenation jointly encode standard visual representation target object representation feed speaker besides concatenation elementwise product compact bilinear pooling also applied training sample triplets speaker listener make word embedding speaker listener shared reduce number parameters reinforcer module sentence sampling using speaker lstm shown top right fig within sampled expressions target objects fed reward function obtain reward values overall loss function formulated learning problem negative matches randomly chosen objects expressions image note listener model limited particular model example computes similarity score every object given referring expression minimizes cross entropy softmax knowing target object could also applied reinforcer besides using pairs target object referring expression training speaker also use reinforcement learning guide speaker toward generating less ambiguous expressions expressions apply target object objects reinforcer module composed discriminative reward function performs policy gradient update speaker specifically given softmax output speaker lstm sample words according categorical distribution time step resulting complete expression sampling end token sampling operation know whether expression ambiguous feed reward function therefore use policy gradient reinforcement learning update speaker parameters goal maximize reward expectation distribution parameterized speaker according policy gradient algorithm log arg min weight reward loss weights loss speaker listener already included eqn eqn list settings sec comprehension generation comprehension task test time could use either speaker listener select target object given input expression using listener would embed input expression learned embedding space select closest object predicted target using speaker would generate expressions object within image select object whose generated log defined softmax output use gradient update speaker model training constrastive pairs set listener objective function eqn set speaker objective function eqn emprically set weight reward loss expression best matches input expression therefore utilize modules ensembling speaker listener predictions together pick probable object given referring expression arg max datasets perform experiments three referring expression datasets refcoco refcocog described sec three datasets collected mscoco images several differences refcoco collected using interactive game interface refcocog collected noninteractive setting contains longer expressions refcocog contains average objects type per images refcoco refcoco average disallowed absolute location words referring expressions overall refcoco expressions objects images expressions objects images refcocog expressions objects images additionally dataset provided dataset splits evaluation refcoco provide person object splits evaluation images containing multiple people testa images containing multiple objects categories testb refcocog authors divide dataset randomly partitioning objects training testing splits thus image may appear two splits training validation splits released dataset use hyperparamters refcoco train models refcocog surprisingly using speaker alone setting already achieves results due joint training adding listener improves performance previous results generation task first let speaker generate multiple expressions per object via beam search use listener rerank expressions select least ambiguous expression similar fully utilize listener power generation propose consider cross comprehension well diversity expressions minimizing potential log log max log first term second term unary potential measure well target object generated expression match using speaker listener modules respectively also used third term unary potential measures likelihood generated sentence describing objects image pairwise potential penalize sentences generated different objects encouraging diversity generation way expressions every object image jointly generated compared previous model attempted tie language generation referring expressions together constraints eqn explicit overall works better reduce ambiguity generated expressions comprehension task training either use speaker listener perform comprehension task speaker models feed every object region within given image speaker select probable region expression comprehension result argmaxoi listener directly compute similarity score expression pick object highest probability evaluation compute iou comprehended region object iou score predicted region greater consider correct comprehension demonstrate benefits module run ablation studies lines table speaker comprehender lines table listener comprehender three datasets row shows results adding module training models speaker listener highlight experiments optimization optimize model using adam initial learning rate halved every iterations batch size word embedding size hidden state size lstm set also set output mlp fig avoid overfitting apply dropout ratio linear transformation mlp layers listener also regularize output layers lstm speaker using dropout ratio listener previous baseline speaker listener previous baseline speaker refcoco val testa testb refcoco detected val testa testb val testa testb detected val testa testb refcocog val refcocog detected val table ablation study using speaker module comprehension task indicated bold top half shows performance given ground truth bounding boxes objects bottom half performance using automatic object detectors select potential objects find adding listener reinforcer modules speaker increases performance listener previous ensemble ensemble listener previous ensemble ensemble refcoco val testa testb refcoco detected val testa testb val testa testb detected val testa testb refcocog val refcocog detected val table ablation study using listener ensembled modules comprehension task indicated bold top half shows performance given ground truth bounding boxes objects bottom half performance using automatic object detectors select potential objects find jointly training speaker improves listener performance adding reinforcer module ensembled prediction performs best module used comprehension bold example means use speaker module joint model comprehension task means use listener module task note speaker module implemented using visdif model mentioned section visdif model trained paper performs slightly better comprehension task original one reported figure example comprehension results based detection green box shows region blue box shows correct comprehension using model red box shows incorrect comprehension use top two rows show correct comprehensions bottom two rows show incorrect ones refcoco test test meteor cider meteor cider test test meteor cider meteor cider refcocog val meteor cider table ablation study generation using automatic evaluation following previous work show results trained mmi trained mmi speakers compare models baseline model line table pure listener model line previous results line achieved els trained mmi outperform mmi also find speaker improved joint training listener module incorporating reinforcer module mmi ranking speaker learned joint training line able outperform pure listener around three datasets already achieves performance comprehension task first evaluate performance speaker comprehension task table observe speaker refcoco test test test test bustly collect expressions objects test sets refcoco obtaining expressions respectively average per object refcocog use original expressions released dataset may limited still show performance completeness choose model reference learns tie expression generation together achieves performance generally find speaker jointly learned models achieves higher scores single speaker metrics across datasets improvements observed settings without addition since previous work found metrics always agree well human judgments referring expressions also run human evaluation set objects refcoco ask turkers click referred object given generated expression results shown table results indicate ablated benefits brought module ultimately joint model achieves best performance fig compare generated expressions using speaker module different models show joint expression generation using full model fig observed expressions every target object considered together meant relevant target object irrelevant objects table human evaluations generation second show evaluations using variations listener module ensembled modules indicated bold comprehension task table note listener generally works better speaker comprehension task indicating deterministic model suitable task speaker model similar results observed reinforcer module seems effective speaker still find joint training always brings additional discriminative benefits listener module resulting improved performance compare line line table ensembling speaker listener together achieves best results overall experiments analyze comprehension performance given ground truth bounding boxes potential comprehension objects algorithm must select objects described provides analysis comprehension performance independent particular object detection method additionally also show results using object detector automatically select regions consideration comprehension bottom half table tables detection algorithm use current state art detector effectiveness speed ssd trained subset coco dataset removing images test splits refcoco validation split refcocog empirically select confidence threshold detection output performance drops somewhat due strong dependence visdif model detection overall improvements brought module consistent using objects showin robustness joint model fig shows comprehension results using full model conclusion demonstrated effectiveness unified framework referring expression generation comprehension model consists speaker listener modules trained jointly improve performance reinforcer module help produce less ambiguous expressions experiments indicate model outperforms state art comprehension generation multiple datasets evaluation metrics references andreas klein reasoning pragmatics neural listeners speakers emnlp antol agrawal mitchell batra lawrence zitnick parikh vqa visual question answering iccv eldon whitney tellex interpreting multimodal referring expressions real time icra fang doering chai embodied collaborative referring expression generation situated interaction proceedings tenth annual international conference interaction generation task generation task evaluate variations speaker module evaluating generation simple comprehension image captioning bleu rouge meteor cider common automatic metrics widely used standard evaluations show automatic evaluation using meteor cider metrics generation table denotes models incorporating reranking mechanism global optimization objects eqn computer cider figure example generation results dataset top bottom showing figure joint generation examples using sentence shows generated expression one depicted objects color coded indicate correspondence fitzgerald artzi zettlemoyer learning distributions logical forms referring expression generation emnlp grice logic conversation cole morgan editors syntax semantics vol speech acts pages academic press san diego grubinger clough deselaers iapr benchmark new evaluation resource visual information systems international workshop ontoimage rohrbach feng saenko darrell natural language object retrieval cvpr jia gavves fernando tuytelaars guiding fukui park yang rohrbach darrell rohrbach multimodal compact bilinear pooling visual question answering visual grounding emnlp golland liang klein approach generating spatial descriptions emnlp poirson yang berg berg modeling context referring expressions eccv term memory model image caption generation iccv johnson karpathy densecap fully convolutional localization networks dense captioning arxiv preprint kazemzadeh ordonez matten berg referitgame referring objects photographs natural scenes emnlp pages kingma adam method stochastic optimization arxiv preprint krahmer van deemter computational generation referring expressions survey computational linguistics krishna zhu groth johnson hata kravitz chen kalantidis shamma visual genome connecting language vision using crowdsourced dense image annotations arxiv preprint lin maire belongie hays perona ramanan zitnick microsoft coco common objects context eccv liu anguelov erhan szegedy reed berg ssd single shot multibox detector eccv mao huang toshev camburu yuille murphy generation comprehension unambiguous object descriptions cvpr mitchell van deemter reiter generating expressions refer visible objects nagaraja morariu davis modeling context objects referring expression understanding eccv plummer wang cervantes caicedo hockenmaier lazebnik entities collecting correspondences richer models iccv rohrbach rohrbach darrell schiele grounding textual phrases images reconstruction eccv simonyan zisserman deep convolutional networks image recognition arxiv preprint vinyals toshev bengio erhan show tell neural image caption generator cvpr wang lazebnik learning deep structurepreserving embeddings cvpr wang azab kojima mihalcea deng structured matching phrase localization eccv williams simple statistical algorithms connectionist reinforcement learning machine learning winograd understanding natural language cognitive psychology park berg berg visual madlibs fill blank description generation question answering iccv
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nonlinear analysis improved swing equation mar pooya claudio nima arjan van der paper investigate properties improved swing equation model synchronous generators model derived omitting main simplifying assumption conventional swing equation requires novel analysis stability frequency regulation consider two scenarios first study case synchronous generator connected constant load second inspect case single machine connected infinite bus simulations verify results ntroduction driven environmental technical motivations restructuring classical power networks vast attention recent decades among goals decreasing energy losses moving towards distributed generation preventing fault propagation building smart microgrids microgrids small power network areas seen single entities large power grids network energy consumption production uncertainty increases great extent according fewer number consumers unpredictable power injections renewables wind turbines solar panels designing controllers electrical sources perturbations abrupt power variations calls rethinking accuracy models mostly valid classical electrical systems main power sources electrical networks synchronous generators despite extensive advances extracting energy renewables machines still main supplier implemented microgrids see unpredictable changes loads renewable generations assure stability frequency regulation grid appears crucial investigate accuracy electrical dynamical models machines large number articles exploited classical swing equation model synchronous generator recently brought doubts accuracy validity proposed models higher accuracy although promising models provided see exact model model use models power networks rather complicated pooya monshizadeh arjan van der schaft johann bernoulli institute mathematics computer science university groningen netherlands paper provide nonlinear analysis improved yet swing equation fulfill higher accuracy analysis modeling synchronous generator also exploited networks first explicate contradicting assumptions obtaining conventional swing equation elaborate deriving improved model section iii first scenario look properties model generator connected constant load provide estimate region attraction nonlinear analysis section second scenario mainly referred single machine infinite bus smib investigated provide analytical estimates region attraction conventional improved swing models case finally simulations provided verification results onventional mproved wing quations mechanical dynamics synchronous generator reads total moment inertia turbine generator rotor rotor shaft velocity mechanical angular velocity associated nominal frequency net mechanical shaft torque counteracting electromagnetic torque coefficient mechanical rotational loss due friction ignored bearing mind model synchronous generator mechanical input electrical output power respectively conventional swing equation approximated constants angular momentum new damping coefficient words swing equation derived supposing assumption contradiction proof stability frequency regulation synchronous generator note machines posses high moment inertia hence claudio persis nima monshizadeh engineering technology institute university groningen netherlands university groningen netherlands despite general misuse accounts damping torque generated damper windings amortisseur generator connected infinite bus case single generator connected load damping term refers proportional torque governor small deviations nominal frequency magnified term dynamics therefore adhere equation endeavor investigate stability frequency regulation nonlinear approach model first suggested however stability synchronous generator connected infinite bus analyzed linearization analysis consistently refer improved swing equation scenario assume injected extracted power constant iii ynchronous enerator onnected onstant oad fig region attraction system estimated stability let equilibrium clearly equation admits real solution standing assumption section assumption obtain following two equilibria system note observe first show locally stable locally unstable moment assume set positive invariant system relax assumption later lyapunov argument provided means well defined interval definition solutions let dynamics rewritten given let given assume holds solutions system starting initial condition set except converge asymptotically equilibrium proof fact stays away zero made clear later according leads using find theorem consider candidate lyapunov function used hence negative set show bearing mind inequality easy check around around hence repulsive following theorem addresses stability equilibrium yields using therefore hence let triple solution left hand side gives show remains positive case together obtain therefore case point belong interval specified cases point lies within boundaries figure illustrates situations since strictly decreasing decreasing well showing stays away time goes proves notice closed bounded invoking lasalle invariance principle solutions system starting asymptotically converge set points equality results completes proof remark note lyapunov function shifted version actual kinetic energy physical system whereas commonly used lyapunov function swing equation equivalent physical interpretation considering mechanical losses add viscous damping coefficient accounting mechanical losses modifying dynamics similarly modified rewritten defining obtain analogous improved swing equation therefore stability results also extends case mechanical losses incremental passivity property facilitate control design section investigate incremental passivity system associated see remark later assume dynamics possesses equilibria little abuse notation following proposition proposition consider candidate lyapunov function assume holds computed along solution satisfies long solution stays within set amounts incremental passivity property respect proof hence inequality holds long solutions evolve set fact follows analogously proof theorem remark defining results dimensionless output consistent inequality incremental passivity property since dimension physical power hold conventional swing equation taken output thus meaningful physical dimension frequency regulation section investigate frequency regulation system integral controller motivated proposition propose controller system fig region attraction system estimated oval characterized asymptotically equilibrium note solutions stay away point since djd rewrite set therefore dissipation equality case reduces long solutions stay within set confines within interval following theorem establishes convergence solutions desired equilibrium associated nominal frequency regulation theorem consider candidate lyapunov function solutions closed loop system starting initial condition set different converge asymptotically equilibrium proof first note ynchronous enerator onnected nfinite scenario consider case smib synchronous generator connected infinite bus inductive link infinite bus node fixed voltage frequency stability improved swing model shown power delivered synchronous generator bus sin voltage angle relative infinite bus voltages reactance line assumed constant dynamics smib system modeled improved swing equation set guarantees remain positive see figure hence find sin long solutions belong set right hand side equality nonpositive conclude set forward invariant along solution thus valid time observe radially unbounded strict minimum invoking lasalle invariance principle solutions converge largest invariant subset trivially satisfies inequality addition solution quantities computed note purpose frequency regulation solution equilibrium interest system given mechanical input considered constant easy check arcsin equilibrium system theorem consider candidate lyapunov function latter shows solutions invariant set starting converge cos cos sin associated kinetic potential energy arcsin assume proof observe minimum convex within set show inequality results reads cos sin inequality contains unique subset precisely criterion arcsin result exists holds figure shows example interval satisfies consequently energy function remains convex remains prove sin sin sin sin sin sin sin sin bearing mind confined set show set leads therefore since thus left hand side inequality shows observe strict minimum solutions bounded invoking lasalle invariance principle solutions converge largest invariant subset set solution satisfy let min solutions system starting initial condition set converge asymptotically equilibrium arcsin arcsin fig estimate system region attraction confines green interval remains positive convex shows solutions invariant set converge asymptotically equilibrium arcsin completes proof remark assumption results arcsin means steady state angle synchronous generator infinite bus less practice steady state angle much lower even large machines see note assumption made merely characterizing region attraction otherwise necessary proof local stability since convex around equilibrium cos comparison swing equation compare results improved model swing equation dynamics synchronous generator modeled swing equation connected infinite bus sin mechanical input considered constant straightforward see equilibrium system arcsin corollary consider candidate lyapunov function cos cos sin assume let solutions smib system described swing equation starting initial condition set converge asymptotically equilibrium arcsin constant load example described section iii steady state value frequency differs swing equation improved swing model similar constant loads figure illustrates issue example set start initial condition steady state value frequency swing equation improved swing model constant load example estimate region attraction improved swing equation provided section iii theorem figure illustrates solution initiates domain becomes unstable however conventional swing equation falsely depicting system remains stable set start according estimate region attraction allows constant load example frequency frequency section provide simulations verifying results examples depict mismatch behavior suggested swing equation improved version simulations parameters set follows note improved swing equation swing equation frequency imulation frequency proof proof similar proof theorem therefore omitted note hence condition suffices proving stability corollary theorem characterize similar region attraction estimates models improved swing conventional one result time fig simulation results case constant load smib show different behavior two models models reach different frequency value bearing load solution improved swing equation starting outside region attraction estimate diverges equilibrium nevertheless solution conventional swing equation converges frequency improved swing equation stable condition hold stability swing equation necessitate condition conventional swing smib model suggests divergence equilibrium frequency improved model converges desired value note initial condition outside region attraction estimate models smib different behavior behavior systems conventional improved similar connected infinite bus however still specific initial conditions systems act quite differently figure illustrates example different behavior angular velocity according analysis section iii condition hold improved swing equation exist steady state frequency value figure shows system becomes unstable inequality violated example adjust defined possesses negative value smib region attraction figure illustrates phase portrait system lyapunov function level sets verified estimate domain attraction conservative note solutions outside estimate region attraction still converge however solutions away estimate domain attraction diverge equilibrium angle rad fig estimate domain attraction system simb verified solutions dotted green within lyapunov level set dark blue converge equilibrium note hence onclusion uture ork investigated properties improved swing equation without relying linearization modeling synchronous generator equation two scenarios considered paper first stability single generator connected constant load proved frequency regulation proposed controller achieved second scenario synchronous machine connected infinite bus contribution respect similar dynamics investigated linearization nonlinear lyapunov analysis provided prove stability frequency regulation finally simulations carried show swing equation model gives rise behavior match suggested improved swing equation future works include considering voltage dynamics systems eferences shahidehpour khodayar cutting campus energy costs hierarchical control economical reliable operation microgrid electrification magazine ieee vol sept zhou ohsawa improved swing equation properties synchronous generators circuits systems regular papers ieee transactions vol jan fiaz zonetti ortega scherpen van der schaft approach power network modeling analysis european journal control vol lagrangian hamiltonian methods modelling control caliskan tabuada compositional transient stability analysis multimachine power networks control network systems ieee transactions vol march caliskan tabuada uses abuses swing equation model decision control cdc ieee annual conference dec natarajan weiss almost global asymptotic stability constant field current synchronous machine connected infinite bus decision control cdc ieee annual conference dec natarajan weiss method proving global stability synchronous generator connected infinite bus electrical electronics engineers israel ieeei ieee convention dec van der schaft stegink perspectives modeling control power networks annual reviews control spring issue appear machowski bialek bumby power system dynamics stability control wiley zhao zheng gao dynamic parameter identification system experimental verification large synchronous machines ieee transactions energy conversion vol sep
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learn fuzz machine learning input fuzzing patrice hila rishabh jan microsoft research risin technion hilap abstract fuzzing consists repeatedly testing application modified fuzzed inputs goal finding security vulnerabilities code paper show automate generation input grammar suitable input fuzzing using sample inputs statistical techniques present detailed case study complex input format namely pdf large complex parser format namely pdf parser embedded microsoft new edge browser discuss measure tension conflicting learning fuzzing goals learning wants capture structure inputs fuzzing wants break structure order cover unexpected code paths find bugs also present new algorithm learn fuzz challenge uses learnt input probability distribution intelligently guide fuzz inputs introduction fuzzing process finding security vulnerabilities code repeatedly testing parser modified fuzzed inputs three main types fuzzing techniques use today blackbox random fuzzing whitebox fuzzing fuzzing viewed variant testing blackbox whitebox fuzzing fully automatic historically proved effective finding security vulnerabilities file parsers contrast fuzzing fully automatic requires input grammar specifying input format application test grammar typically written hand process laborious time consuming nevertheless fuzzing effective fuzzing technique known today fuzzing applications complex structured input formats like must take untrusted inputs including complex html documents javascript code paper consider problem automatically generating input grammars fuzzing using techniques work author done mostly visiting microsoft research obj pages kids dobj xref trailer root startxref fig excerpts pdf document sample object table one subsection trailer sample inputs previous attempts used variants traditional automata learning algorithms see section contrast prior work paper presents first attempt using statistical learning techniques problem specifically use recurrent neural networks learning statistical input model also generative used generate new inputs based probability distribution learnt model see section introduction learning techniques use unsupervised learning approach fully automatic require customization present case study complex input format pdf format complex see section described pdf document consider large complex parser format pdf parser embedded microsoft new edge browser series detailed experiments see section discuss learn fuzz challenge learn generate diverse inputs order maximize coverage still injecting enough input parts order exercise unexpected code paths code also present novel learn fuzz algorithm section uses learnt input probability distribution intelligently guide fuzz statistically inputs show new algorithm outperform random fuzzing algorithms considered work paper organized follows section presents overview pdf format specific scope work section gives brief introduction learning discusses use adapt techniques learn fuzz problem section presents results several learning fuzzing experiments edge pdf parser related work discussed section conclude discuss directions future work section end obj work dobj dobj end obj fig pdf data objects different types structure pdf documents full specification pdf format pages long specification roughly deals description data objects relationships parts pdf document pdf files encoded textual format may contain binary information streams images encrypted data pdf document sequence least one pdf body pdf body composed three sections objects table trailer objects data metadata pdf document organized basic units called objects objects similarly formatted seen figure joint outer structure first line object identifier indirect references generation number incremented object overridden newer version obj indicates start object endobj indicator closes object object figure contains dictionary structure delimited contains keys begin followed values reference object document identifier generation number since document large referenced object accessed using via table examples objects shown figure object figure content array object purpose hold coordinates referenced another object figure string literal holds bookmark text pdf document section figure numeric object figure object containing array examples object types used basic blocks objects composed dictionary object figure contains array rules defining composing objects comprises majority specification cross reference table cross reference tables pdf body contain address bytes referenced objects within document figure shows table subsection contains addresses five objects identifiers placeholder identifier never refers object object pointed determined row table subsection include objects starting identifier indicator object use first column address object file object used first column refers identifier previous free object case object object last available object closing circle trailer trailer pdf body contains dictionary contained within information body startxref address table allows body parsed end reading startxref skipping back table parsing parsing objects needed updating document pdf documents updated incrementally means pdf writer wishes update data object start new pdf body write new object identifier generation number greater one appeared write new table pointing new object append body previous document similarly object deleted creating new table marking free use method order append new objects pdf file discussed later section scope work paper investigate leverage adapt learning techniques learn grammar pdf data objects data objects formatted text shown figure figure rules defining composing data objects makes bulk specification rules numerous tedious repetitive structured therefore learning neural networks show later contrast learning automatically structure rules defining tables trailers involve constraints lists addresses pointers counters look complex less promising learning neural networks also consider binary data objects encoded binary image blackbox whitebox fuzzing already effective statistical learning object contents describe statistical learning approach learning generative model pdf objects main idea learn generative language model set pdf object characters given large corpus objects use network model shown produce results many different learning tasks machine translation speech recognition model allows learning arbitrary length contexts predict next sequence characters compared traditional based approaches limited contexts finite length given corpus pdf objects model trained unsupervised manner learn generative model generate new pdf objects using set input output sequences input sequences correspond sequences characters pdf objects corresponding output sequences obtained shifting input sequences one position learnt model used generate new sequences pdf objects sampling distribution given starting prefix obj neural network models recurrent neural network rnn neural network operates variable length input sequence consists hidden state output rnn processes input sequence series time stamps one element sequence given time stamp hidden state time stamp output computed activation function sigmoid tanh etc function softmax computes output probability distribution given vocabulary conditioned current hidden state rnns learn probability distribution character sequence training predict next character sequence learn conditional distribution cho introduced model consists two recurrent neural networks encoder rnn processes variable dimensional input sequence fixed dimensional representation decoder rnn takes fixed dimensional input sequence representation generates variable dimensional output sequence decoder network generates output sequences using predicted output character generated time step input character timestep illustration architecture shown figure architecture allows learn conditional distribution sequence next outputs train model using corpus pdf objects treating one sequence characters training first concatenate object files single file resulting large sequence characters split sequence multiple training sequences fixed size ith training instance denotes subsequence indices output sequence training sequence input sequence shifted position model trained learn generative model set training instances encoder rnn decoder rnn fig rnn model generate pdf objects generating new pdf objects use learnt model generate new pdf objects many different strategies object generation depending upon sampling strategy used sample learnt distribution always start prefix sequence obj denoting start object instance query model generate sequence output characters produces endobj corresponding end object instance describe three different sampling strategies employ generating new object instances nosample generation strategy use learnt distribution greedily predict best character given prefix strategy results generating pdf objects likely consistent also limits number objects generated given prefix like obj best sequence next characters uniquely determined therefore strategy results pdf object limitation precludes strategy useful fuzzing sample generation strategy use learnt distribution sample next characters instead selecting top predicted character sequence given prefix sequence sampling strategy able generate diverse set new pdf objects combining various patterns model learnt diverse set objects training corpus sampling generated pdf objects always guaranteed useful fuzzing perspective samplespace sampling strategy combination sample nosample strategies samples distribution generate next character current prefix sequence ends whitespace whereas uses best character distribution middle tokens prefixes ending characters similar nosample strategy strategy expected generate pdf objects compared sample strategy sampling restricted end whitespace characters algorithm samplefuzz tfuzz seq obj endobj sample seq sample learnt distribution pfuzz random random variable decide whether fuzz pfuzz tfuzz seq replace lowest likelihood end seq seq len seq maxlen seq obj reset sequence end end return seq samplefuzz sampling fuzzing goal learning generative model pdf objects ultimately perform fuzzing perfect learning technique would always generate objects would exercise code whereas bad learning technique would result objects woult quickly rejected parser upfront explore tradeoff present new algorithm dubbed samplefuzz perform fuzzing sampling new objects use learnt model generate new pdf object instances time introduce anomalies exercise code samplefuzz algorithm shown algorithm takes input learnt distribution probability fuzzing character tfuzz threshold probability used decide whether modify predicted character generating output sequence seq algorithm samples learnt model get next character probability particular timestamp probability higher threshold model confident likely next character sequence algorithm chooses instead sample another different character place minimum probability learnt distribution modification fuzzing takes place result pfuzz random coin toss returns probability higher input parameter tfuzz lets user control probability fuzzing characters key intuition samplefuzz algorithm introduce unexpected characters objects places model highly confident order trick pdf parser algorithm also ensures object length bounded maxlen note algorithm guaranteed always terminate observe always terminates practice training model since train model unsupervised learning setting test labels explicitly determine well learnt models performing instead train multiple models parameterized number passes called epochs learning algorithm performs training dataset epoch thus defined iteration learning algorithm complete training dataset evaluate models trained five different numbers epochs setting one epoch takes minutes train model model epochs takes hours learn use lstm model variant rnn hidden layers layer consists hidden states experimental evaluation experiment setup section present results various fuzzing experiments pdf viewer included microsoft new edge browser used singleprocess executable provided windows team purposes executable takes pdf file input argument executes pdf parser included microsoft edge browser stops executable detects parsing error due pdf input file malformed prints error message execution log follows simply refer edge pdf parser experiments performed windows vms ram use three main standard metrics measure fuzzing effectiveness coverage test execution measure instruction coverage set unique instructions executed test instruction uniquely identified pair values coverage set tests simply union coverage sets individual test pass rate test execution programmatically check grep presence messages execution log error messages call test pass otherwise call fail pass tests corresponds pdf files considered edge pdf parser metric less important fuzzing purposes help estimate quality learning bugs test execution performed monitoring tool appverifier free runtime monitoring tool catch memory corruptions bugs buffer overflows low runtime overhead typically percent runtime overhead widely used fuzzing windows instance sage detects bugs training data extracted pdf objects diverse set pdf files files provided windows fuzzing team used prior extended fuzzing edge pdf parser set files result seed minimization process computing subset larger set input files provides instruction coverage larger set seed minimization standard first step applied file fuzzing larger set pdf files came various sources like past pdf files used fuzzing also pdf files collected public web objects training set rnns used work binary objects embedded pdf files typically representing images various image formats considered work learn generate fuzz pdf objects edge pdf parser processes full pdf files single objects therefore wrote simple program correctly append new pdf object existing pdf file call host following procedure discussed section updating pdf document specifically program first identifies last trailer pdf host file provides information file addresses objects table last used object next new body section added file new object included object overrides last object host file new cross reference table appended increases generation number overridden object finally new trailer appended baseline coverage allow meaningful interpretation coverage results randomly selected pdf objects training objects measured coverage edge pdf parser used baseline later experiments first question host pdf file use experiments since pdf file objects new appended object interfere objects already present host hence influence overall coverage pass rate study question selected smallest three pdf files set files used hosts three hosts size respectively figure shows instruction coverage obtained running edge pdf parser three hosts denoted also show coverage obtained computing union three sets denoted coverage ranges unique instructions union larger three host covers unique instructions covered two note smallest file lead smallest coverage fig coverage pdf hosts baselines next recombined baseline objects three hosts obtain three sets new pdf files denoted respectively figure shows coverage set well union observe following baseline coverage varies depending host larger host alone expected largest difference host baseline coverage instruction instruction words instructions included host coverage matter new objects appended test typically covers order half million unique instructions confirms edge pdf parser large application pdf files take minutes processed tested get coverage data also measured pass rate experiment expected pass rate hosts main takeaway even though coverage varies across hosts objects may interact differently host pdf file always perceived edge pdf parser learning pdf objects training rnn important parameter number epochs used see section report results experiments obtained training rnn epochs respectively training used learnt rnn model generate unique pdf objects also compared generated objects objects used training model found exact matches explained earlier section consider two main rnn generation modes sample mode sample distribution every character fig pass rate sample samplespace epochs position samplespace mode sample distribution whitespaces generate top predicted character positions pass rate sample samplespace training epochs reported figure observe following pass rate samplespace consistently better one sample epochs pass rate sample already means learning good quality number epochs increases pass rate increases expected since learned models become precise also take time see section best pass rate obtained samplespace epochs interestingly pass rate essentially regardless host pdf file used varies across hosts data shown main takeaway pass rate ranges shows learning good quality coverage learned pdf objects figure shows instruction coverage obtained sample samplespace epochs using top left top right bottom left overall coverage hosts bottom right figure also shows coverage obtained corresponding baseline observe following unlike pass rate host impacts coverage significantly already pointed earlier moreover shapes line vary across hosts coverage sample samplespace baseline coverage epoch results mostly baseline coverage fig coverage sample samplespace epochs host best overall coverage obtained sample see data bottom right coverage overall second best behind sample best coverage obtained samplespace also main takeaway best overall coverage obtained sample comparing coverage sets far simply counted number unique instructions covered drill overall coverage data figure compute overlap overall coverage sets obtained winner well overall coverage results presented figure observe following sets almost supersets expected see row except hundred instructions almost superset sets except instructions compared hundreds instructions compared see column fig comparing coverage unique instructions row compared column algorithm samplefuzz coverage pass rate fig results fuzzing experiments pdf files way instructions common differ better coverage incomparable instructions also missing instructions main takeaway coverage winner almost superset coverage sets combining learning fuzzing section consider several ways combine learning fuzzing evaluate effectiveness consider simple blackbox random fuzzing algorithm denoted random randomly picks position file replaces byte value random value algorithm uses fuzzfactor length file divided average number bytes fuzzed file use random generate variants every pdf object generated baseline resulting fuzzed objects host files obtain three sets new pdf files denoted respectively comparison purposes also include results running generate objects denoted finally consider new algorithm samplefuzz described section decides fuzz values based learnt distribution applied algorithm learnt distribution rnn model tfuzz threshold figure reports overall coverage set set pdf files takes hours processed rows sorted increasing coverage observe following applying random objects generated sample samplespace baseline coverage goes pass rate goes consistently analyzing overlap among coverage sets data shown fuzzed sets almost supersets original sets expected coverage also increases instructions compared sample pass rate remains around expected perhaps surprisingly best overall coverage obtained samplefuzz pass rate difference absolute coverage samplefuzz next best instructions moreover analyzing coverage set overlap samplefuzz covers instructions also misses instructions covered therefore none two winners fully simulate effects main takeaway algorithms considered competitive compared three beat baseline coverage main takeaway tension coverage pass rate main takeaway experiments tension observe coverage pass rate tension visible figure also visible earlier results correlate coverage results figure results figure clearly see samplespace better pass rate sample sample better overall coverage samplespace see bottom right figure intuitively tension explained follows pure learning algorithm like samplespace generates almost objects exercises little code contrast noisier learning algorithm like sample lower generate many objects also generates ones exercise code applying random fuzzing algorithm like random nearly objects even dramatic effect lowering pass rate see figure increasing coverage probably due increased coverage code new samplefuzz algorithm seems hit sweet spot pass rate coverage experiments sweet spot pass rate seems around pass rate high enough generate diverse wellformed objects cover lot code pdf parser yet low enough also exercise code many parts parser note instruction coverage ultimately better indicator fuzzing effectiveness pass rate instead metric bugs addition coverage pass rate third metric interest course number bugs found experiments previously reported section bugs found note edge pdf parser thoroughly fuzzed months fuzzers including sage performed study bugs found prior fuzzing fixed version pdf parser used study however longer experiment objects pdf files took nearly days bug found edge pdf parser pdf file generated size triggers unexpected recursion parser ultimately results stack overflow bug later confirmed fixed microsoft edge development team plan conduct longer experiments near future related work fuzzing popular blackbox random fuzzers today support form grammar representation among many others work test input generation started related testing test generation grammar usually either random exaustive imperative generation related approach program generates inputs effect program encodes grammar fuzzing also combined whitebox fuzzing learning grammars fuzzing bastani present algorithm synthesize grammar given set input examples used generate new inputs fuzzing algorithm uses set generalization steps introducing repetition alternation constructs regular expressions merging grammars turn results monotonic generalization input language technique able capture hierarchical properties input formats well suited formats pdf objects relatively flat include large diverse set content types pairs instead approach uses models learn statistical generative http http models flat formats moreover learning statistical model also allows guiding additional fuzzing generated inputs autogram also learns grammars given set inputs dynamically observing inputs processed program instruments program test dynamic taints tags memory input fragments come parts inputs processed program become syntactic entities grammar tupni another system reverse engineers input format examples using taint tracking mechanism associate data structures addresses application address space unlike approach treats program test autogram tupni require access program adding instrumentation complex applicability precision complex formats pdf objects unclear program analysis lot recent interest using neural networks program analysis synthesis several neural architectures proposed learn simple algorithms array sorting copying neural flashfill uses novel neural architectures encoding examples generating programs domain specific language several based models developed learning repair syntax errors programs techniques learn model set correct programs use learnt model predict syntax corrections buggy programs related work optimizes assembly programs using neural representations paper present novel application models learn grammars sample inputs fuzzing purposes conclusion future work fuzzing effective fuzzing applications complex structured inputs provided comprehensive input grammar available paper describes first attempt using statistical learning techniques automatically generate input grammars sample inputs presented evaluated algorithms leverage recent advances sequence learning neural networks namely recurrent neural networks automatically learn generative model pdf objects devised several sampling techniques generate new pdf objects learnt distribution show learnt models able generate large set new wellformed objects also results increased coverage pdf parser used experiments compared various forms random fuzzing results presented section may vary applications general observations tension conflicting learning fuzzing goals remain valid learning wants capture structure inputs fuzzing wants break structure order cover unexpected code paths find bugs believe inherent statistical nature learning neural networks powerful tool address learn fuzz challenge several interesting directions future work focus paper learning structure pdf objects would worth exploring learn automatically possible hierarchical structure pdf documents involving tables object bodies trailer sections maintain certain complex invariants amongst perhaps combination logical inference techniques neural networks could powerful enough achieve also learning algorithm currently agnostic application test considering using form reinforcement learning guide learning models coverage feedback application could potentially guide learning explicitly towards increasing coverage acknowledgments thank dustin duran mark wodrich microsoft windows security team helpful feedback also thank team members project springfield partly funded work references adobe systems incorporated pdf reference edition available http osbert bastani rahul sharma alex aiken percy liang synthesizing program input grammars corr sahil bhatia rishabh singh automated correction syntax errors programming assignments using recurrent neural networks corr rudy bunel alban desmaison pawan kumar mudigonda pushmeet kohli philip torr adaptive neural compilation nips pages kyunghyun cho bart van merrienboer dzmitry bahdanau fethi bougares holger schwenk yoshua bengio learning phrase representations using rnn statistical machine translation emnlp pages jan chorowski dzmitry bahdanau dmitriy serdyuk kyunghyun cho yoshua bengio models speech recognition advances neural information processing systems pages claessen hughes quickcheck lightweight tool random testing haskell programs proceedings icfp coppit lian yagg generator structured test inputs ase weidong cui marcus peinado karl chen helen wang luis tupni automatic reverse engineering input formats proceedings acm conference computer communications security pages acm brett daniel danny dig kely garcia darko marinov automated testing refactoring engines fse godefroid kiezun levin whitebox fuzzing proceedings pldi acm sigplan conference programming language design implementation pages tucson june godefroid levin molnar automated whitebox fuzz testing proceedings ndss network distributed systems security pages san diego february rahul gupta soham pal aditya kanade shirish shevade deepfix fixing common language errors deep learning aaai hanford automatic generation test cases ibm systems journal sepp hochreiter schmidhuber long memory neural computation matthias andreas zeller mining input grammars dynamic taints ase pages karol kurach marcin andrychowicz ilya sutskever neural machines arxiv preprint schulte controllable combinatorial coverage testing testcom majumdar directed test generation using symbolic grammars ase maurer generating test data enhanced grammars ieee software emilio parisotto mohamed rishabh singh lihong dengyong zhou pushmeet kohli program synthesis corr yewen karthik narasimhan armando regina barzilay neural program corrector moocs corr purdom sentence generator testing parsers bit numerical mathematics scott reed nando freitas neural arxiv preprint sirer bershad using production grammars software testing dsl ilya sutskever oriol vinyals quoc sequence sequence learning neural networks advances neural information processing systems pages sutton greene amini fuzzing brute force vulnerability discovery utting pretschner legeard taxonomy testing department computer science university waikato new zealand tech rep
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rairo operations research jun set publisher complexity approximation results scheduling problem according topology darties giroudeau simonin abstract consider makespan minimization problem presence compatibility constraints specified topology particular focus stretched execution time idle time duration study several problems framework classic complexity approximation compatibility graph bipartite star chain context design efficient approximation algorithms intractable scheduling problem according parameters introduction model detection object common radar system based following principle transmitter emits pulse propagates though environmental medium pulse encounters object reflected back transmitter using transmit time direction pulse position object computed transmitter formally acquisition process divided three parts pulse transmission wave propagation reflection iii echo reception thus detection system must perform two date precede ams documentclasses reported umr cnrs university burgundy rue alain savary dijon france lirmm umr cnrs montpellier rue ada montpellier cedex france lirmm umr cnrs montpellier rue ada montpellier cedex france insight centre data analytics university college cork ireland edp sciences title set publisher tasks parts iii separated idle time part systems generally run mode started task interrupted resumed later however idle time acquisition task reused perform another task systems scheduling issues appear parallel several sensors using different frequencies working idle time acquisition task reused perform partially entirely second acquisition process using another sensor sensors use different frequencies avoid interferences otherwise two acquisitions processes scheduled sequentially introduced first shapiro seem natural way model among others data acquisition processes composed two processing time whose execution must separated incompressible flexible time called idle time task acquisition process sensor emits radio pulse first subtask listens echo reply second radio pulse propagation operates idle time also efficient way model acquisition systems designed detect changes environment given period producing two measurements given period measurement modeled note collection scheduled order minimize makespan schedule length necessary execute one several different idle time original model may executed according processing time duration idle time papers investigated problem minimizing makespan various configurations depending values authors present global visualization scheduling problems complexity coupledtasks give main complexity results acquisition system incompatibilities may arise two tasks operate two different sensors working channel thus valid schedule would require scheduled sequentially hereafter propose generalization original model considering notion compatibility constraint among tasks original model introducing compatibility constraint among tasks two tasks compatibles may executed idle time reciprocally introduced compatibility graph model compatibility entire collection pair compatible tasks linked edge proposed new results focused impact addition complexity problem work motivated acquisition data automatic vehicle water taipan torpedo growth robotic technologies several applications works emerging theoretical needs priority example torpedo used execute several submarine topographic surveys title set publisher including topological temperature measurements acquisitions tasks partitioned specific modelling precise since engineers wide degree freedom create transform different tasks required strong theoretical analysis coupled tasks compatibility constraint indeed needed better knowledge difficulty scheduling systems compare scheduling heuristics optimal one contribution work propose new results complexity approximation particular problem instances composed stretched stretched three parameters equal value called stretch factor investigate problem scheduling set stretched subject compatibility constraint order minimize completion time latest task clarity resp refers either first resp second processing time according context major research issue concerns impact class compatibility graph complexity problem known problem phard even tasks compatibles complete graph see side empty graph trivial optimal solution would consist scheduling tasks sequentially aim determine complexity problem describes bipartite graphs propose approximation algorithms performance guarantee instances remark two compatibles stretched scheduled parallel solution scheduling problem one following conditions must hold either idle time one task fully exploited schedule scheduled scheduled execution two tasks done without idle time fully executed idle time sake simplify say pack others configuration unavailable otherwise would overlap schedule remark one propose orientation edge task lowest stretch factor task highest one set bidirectional edge following consider oriented compatibility graphs abusing notation dealing undirected topologies refers fact underlying undirected graph use various standard notations graph theory set neighbors maximum degree denote respectively title set publisher indegree outdegree denote graph induced vertices reusing graham notation scheme define main problem study study variation complexity topology varies propose approximation results instances study thee subclasses bipartite graphs particular chain star bipartite graphs bipartite graph graph arc extremities given bipartite graph denote kth stage paper focus study bipartite graphs bipartite graphs also study problem compatibility graph complete bipartite graph contains edges denoted resp set tasks represented resp set let seq time required schedule sequentially tasks formally seq remark given instance independent set tasks pairwise thus seq lower bound cost optimal solution results obtained article summarized table topology graph graph graph complexity theo theo theo theo approximation theo theo theo theo theo table complexity approximation results discussed paper see prerequisites performance ratio recall performance ratio minimization resp maximization problem given ratio value approximation solution rea turned algorithm instance optimum maxi opt graph incoming arcs central node arc graph least one outcoming arc central node title set publisher resp mini opt notice minimization problem ratio greater one resp lower one definition problems prove different results announced paper use several approximation results four problems subset sum problem problem given set positive values one asks exists subset decision problem pcomplete see optimization version problem sometimes viewed knapsack problem item profits weights coincide value knapsack capacity aim find set packable items maximum profit multiple subset sum mss problem variant bin packing number identical bins given one would like maximize overall weight items packed bins sum item weights every bin exceed bin capacity problem also special case multiple knapsack problem knapsacks capacities item profits weights coincide caprara proved mss admits admit even two knapsacks also proposed algorithm multiple subset sum different knapsack capacities mssdc extension mss considering different bin capacities mssdc also admits generalization mssdc multiple knapsack assignment restriction mkar problem consists pack weighted items nonidentical bins additional constraint item packed bins item profit objective maximize sum profits packed items considering profit item equals weight proposed approximation also use result concerning variant problem denoted subsequently sat instance sat described following elements use denote set variables let multiple let set clauses cardinality clauses cardinality clauses cardinality clause cardinality equal literals resp literals belongs one clauses cardinality thus one positive literals belongs one clauses cardinality thus one title set publisher whenever clause cardinality belong different clauses cardinality aim sat find exists truth assignment false true whereby clause exactly one true literal sat proven example following logic formula smallest valid instance sat answer sat yes sufficient choose yields truth assignment satisfies formula exactly one true literal clause computational complexity classes compatibility graphs section present two preliminary results complexity problem consists scheduling set tasks compatibility constraints context consider topologies chain star first show problem solvable within time complexity algorithm chain theorem prove even compatibility graph star theorem chain graph despite simplicity chain topology solving scheduling problem chain simple appears main issue arise two adjacent vertices stretch factor configuration determine locally packed together optimal solution requires examine neighbourhood problematic configuration repeated along chain however show scheduling problem chain polynomial using similar method developed theorem problem admits algorithm proof problem solved reduction search minimum weighted perfect matching problem polynomially solved time complexity first note task two neighbors idle duration high enough schedule thus one schedule without decreasing cost optimal solution remove tasks studied graph thus rest proof one restrict study chains title set publisher order obtain graph even number vertices find perfect matching construct graph define weighted function follows let instance problem compatibility graph instance minimum weight perfect matching problem graph constructed consider graph consisting two copies denoted vertex corresponding denoted moreover edge added state weight represents sequential time even size two compatible tasks add edges state two compatible tasks add edges state figure shows example construction chain vertices figure example transformation one show weighted perfect matching cover vertices fact construction implies perfect matching cost one provide valid schedule processing time scheduling problem edge implies task scheduled alone edge implies tasks packed together resulting schedule edge belong also matching minimum weight perfect matching associate schedule minimum processing time equal vice versa detailed proof relationship solution problem solution minimum weight perfect matching presented title set publisher review literature edmonds algorithm determines minimum weight perfect matching optimization problem polynomial one adds execution blocks created removed vertices leads polynomiality problem star graph focus case star graph graph central node context show complexity depends number outgoing arcs following results also imply studied problem phard even acyclic graphs degree unbounded theorem problem polynomial central node admits least one outcoming arc proof let set satellite nodes according remark seq lower bound cost optimal solution bound achieved execute central node satellite node theorem problem central node admits incoming arcs proof propose reduction subset sum problem see section instance composed set positive values construct instance following way value introduce let set tasks add task define compatibility constraint task clearly compatibility graph star central node transformation computed polynomial time prove exists positive solution subset sum problem iff exists feasible solution scheduling problem length easy see let set nodes executed central node scheduling cost scheduling clearly seq seq therefore problem finding scheduling cost seq clearly equivalent instance subset sum set processing time satellite tasks concludes proof theorem title set publisher boundary algorithm bipartite graphs preliminary results section show problem acyclic instances degree unbounded suggest complexity problem may linked maximum degree graph section devoted several scheduling problems presence bipartite compatibility graph according maximum degree vertices structural parameters like number different values sharp line demarcation polynomially solvable cases ones according several topologies focus analysis bipartite graph prove problem solvable within polynomial algorithm theorem becomes theorem start designing algorithm scheduling problem maximum degree incoming arcs two theorem problem deciding whether instance stagebipartite polynomial fact previous result may extended graph necessarily bipartite neighbors proof let bipartite compatibility graph arcs oriented implying executed idle time always scheduled sequentially independent set remark aim fill idle time maximum remained tasks executed minimize length remained tasks easy see incoming degree equal one executed sequentially idle time following algorithm focused case defined two steps task two neighbors add schedule execute sequentially idle time remove instance remaining task admits two incoming arcs add weight arc perform maximum weight matching order minimize length remained tasks thus matched executed tasks removed remaining tasks processed sequentially tasks complexity algorithm using hungarian method extension sufficient use maximum weight perfect matching title set publisher theorem problem deciding whether instance stage bipartite schedule length number tasks proof easy see problem proof based reduction sat given set boolean variables mod set clauses cardinality two clauses cardinality three construct instance problem bipartite following way introduce four set noted introduce three set denoted clauses length three introduce two clauses length two introduce one set denoted following arcs model compatibility constraint boolean variables add arcs clauses length three denoted add arcs clauses length two denoted add arcs finally add arcs transformation clearly computed polynomial time illustration depicted figure proposed compatibility graph bipartite follows say task merged task exists compatibility constraint may executed idle time let first assume schedule length prove truth assignment clause exactly one true literal one literal equal make several essential remarks length schedule given execution time admitting incoming arcs value length thus tasks must merged tasks construction three tasks merged together merged title set publisher case true false two length two case false true two length two figure partial compatibility graph scheduling problem bipartite title set publisher allocation leads idle time length equals idle times merged simultaneously one tasks merged previous discussion necessary merge one impossible size way merged simultaneously hence first merged variable occurs second merged affect value true variable iff merged corresponding clause variable occurs obvious see clause length three two one one literal equal true conversely suppose truth assignment clause exactly one true literal variable true merged vertices corresponding clause length three occurs corresponding clause length two occurs variable alse merged vertices corresponding clause length two occurs corresponding clause length three occurs given figure feasible schedule sufficient merge vertices partition thus length schedule theorem problem deciding whether instance stage bipartite schedule length number tasks proof use similar proof given theorem sufficient add clause length two resp length three dummy resp value resp words add two compatibility constraints clause length two clause length three schedule length iff exists truth assignment clause exactly one true literal one literal equal title set publisher corollary problem deciding whether instance stage schedule length number tasks proof proof theorem adapted using classical scaling arguments assigning task approximation algorithms section devoted design efficient approximation algorithms several topologies mainly bipartite graphs authors proposed simple algorithm consists scheduling tasks consecutively performance ratio bounded general compatibility graph challenge remaining section propose efficient algorithms ratio strictly lower propose star graph whereas apx developed remaining section according characteristics bipartite graph last extend result extended bipartite graph star graph theorem problem admits proof central node admits incoming arcs case central node admits least one outcoming arc given corollary therefore may use solution given subset sum see indeed based reduction used proof theorem optimization version aim find optimal set tasks executed idle time central node maximizes seq seq seq let suppose seq designates value approximation solution subset sum note seq seq lead seq seq seq seq seq seq seq seq seq seq seq seq seq therefore existence subset sum involves scheduling problem title set publisher bipartite graph scheduling idle time others related packing problems especially compatibility graph bipartite graph following propose several approximations bipartite graph lemma let problem mkar mssdc mss admits following problems stage complete stage complete stage tasks stretch factor posses within factor using approximability reduction mkar mssdc mss respectively proof consider instance stage graph arc stretch factor function instance valid schedule consists finding task subset compatible tasks pack task packed let union packed task let set remaining tasks obviously seq seq seq independent set scheduled sequentially optimal solution length schedule time required execute entirely plus one required schedule sequentially tasks formally cmax seq seq equations cmax seq seq seq use transformation mkar problem task item weight task bin capacity item packed edge belongs bipartite graph using algorithms results literature one obtain assignment items bins denote set packed items cost solution mkar problem seq title set publisher mkar approximable factor seq seq set packable items maximum profit combining equations obtain solution stage length cmax seq seq seq two fixed sets optimal solution minimal length cmax obtained seq maximum length optimal solution equal cmax seq seq seq using equations ratio obtained solution optimal one seq seq seq seq cmax cmax seq seq seq seq seq seq definition moreover processing time excess idle time tasks obtain seq seq combined equation obtain desired upper bound cmax cmax problem complete proof similar previous one remind mssdc special case mkar item packed bin problem complete stage stretch factor proof similar previous one since mssdc generalization mss theorem following problems admit approximation algorithm title set publisher problem stage approximable factor problem complete stage admits problem complete stage stretch factor approximable factor admits proof authors proposed algorithm mkar reusing result lemma obtain stage know mssdc admits using algorithm compute lemma obtain approximation ratio problem case two different results authors proposed algorithm mss reusing result lemma obtain complete also proved mss admits using algorithm compute lemma obtain approximation ratio complete stage stretch factor bipartite graph following extend previous result bipartite graphs theorem problem stage approximable factor proof main idea algorithm divided three steps first compute part solution thanks theorem stage compute second part solution thanks theorem stage finish merge two parts resolve potential conflicts giving preference tasks packed computing cost solution gives approximation ratio reusing notation introduced bipartite graphs see section consider instance stage arc extremities title set publisher resp set tasks merged resp remaining task solution moreover let vib set tasks scheduled tasks merged notation extended optimal solution adding star involved variables considering specificities instance optimal solution propose essential remarks tasks source nodes either scheduled alone merged tasks given solution problem partition tasks either scheduled alone merged tasks scheduled tasks merged given solution problem partition tasks form independent set scheduled sequentially schedule remark either alone tasks merged given solution problem partition solution would consist first schedule task least one task merged schedule remaining tasks alone consecutively given optimal solution length given following equation seq seq seq seq seq simply seq seq seq seq note part equation scheduled idle time main idea algorithm divided three steps first compute part solution thanks theorem stage compute second part solution thanks theorem stage finish merge two parts solve potential conflicts let consider instance restricted graph note bipartite graph let optimal solution let denote set tasks packed tasks set remaining tasks obviously seq packed alone box title set publisher given solution let set tasks least one task merged remaining tasks let denote set tasks merged set remaining tasks using theorem lemma demonstration presented proof compute solution seq seq note seq seq seq seq seq combining equations obtain seq seq seq seq seq know definition seq seq use similar reasoning instance restricted graph let optimal solution let denote set tasks packed tasks set remaining tasks given solution let set tasks least one task merged remaining tasks one compute solution based set tasks packed seq seq seq seq seq know definition seq seq design feasible solution follows first compute solution add task tasks merged second compute solution add task tasks merged third tasks added scheduled sequentially last denote vconf lict set remaining tasks set tasks merged task thus merged task observe seq seq title set publisher thus cost solution seq seq seq seq vconf lict also clear seq vconf lict seq seq using equations equation obtain seq seq seq seq seq using equations obtain seq seq know seq tasks must merged tasks exceed idle time implying seq seq write following seq seq seq know tasks must merged tasks exceed idle time implying seq seq also know seq independent set one write following seq seq seq finally equations conclude proof conclusion paper investigate particular scheduling problem presence compatibility graph regard complexity approximation also establish specific case bipartite compatibility graph context title set publisher propose algorithm bound tight extend result bipartite designing interesting question concerns study complexity tree graphs bounded degree known complexity result exists another perspective consists extending presented results bipartite graphs references ahr galambos oswald reinelt exact algorithm scheduling identical mathematical methods operations research june ecker kis potts tanas whitehead scheduling coupled tasks unit processing times technical report poznan university technology caprara kellerer pferschy ptas multiple subset sum problem different knapsack capacities inf process lett caprara kellerer pferschy multiple subset sum problem siam journal optimization caprara kellerer pferschy algorithm multiple subset sum heuristics dawande kalagnanam keskinocak salman ravi approximation algorithms multiple knapsack problem assignment restrictions journal combinatorial optimization edmonds maximum matching polyhedron vertices journal research national bureau standards garey johnson computers intractability guide theory freeman new york usa giroudeau palaysi complexity approximation precedence constrained scheduling problem large communication delays theoretical computer science graham lawler lenstra rinnooy optimization approximation deterministic sequencing scheduling survey annals discrete mathematics ibarra kim fast approximation algorithms knapsack sum subset problems journal acm kellerer mansini pferschy speranza efficient fully polynomial approximation scheme problem journal computer system sciences kuhn hungarian method assignment problem naval research logistics quarterly orman potts complexity scheduling discrete applied mathematics shapiro scheduling coupled tasks naval research logistics quarterly simonin darties giroudeau isomorphic scheduling problem compatibility constraints single processor journal scheduling simonin giroudeau complexity approximation scheduling problem presence compatibility tasks project management scheduling simonin giroudeau darties theoretical aspects scheduling presence compatibility graph algorithmic operations research
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apr ori parzanchevski peter sarnak david kazhdan admiration abstract symmetry groups platonic solids adjoin carefully designed involution yielding topological generators optimal covering properties well efficient navigation consequence optimal strong approximation integral quadratic forms associated certain special quaternion algebras arithmetic groups generators give super efficient quantum gates natural building blocks design universal quantum gates introduction circuits used quantum computation unitaries products elementary unitaries operates fixed typically number qubits standard universal gate set quantum computing consists unitaries applied single qubits xor gate together generate classical computer operations single bit leave flip quantum setting rotate unitaries reduce finite universal gate set one settle topologically dense set since overall phases matter suffices find good topological generators algorithm ensures fixed topological generators reasonably short words circuits approximate respect metric point view general polynomial type complexity classes sufficient however much interest theoretical practical optimize choice generators correspond special arithmetic subgroups unit quaternions introduced optimal generating rotations yield variants optimal generators particular case clifford plus gates described appear textbooks clifford gates form finite subgroup order make set universal one add extra element popular choice order element generate group shown see also considerations applying make circuits require among things universal gate set consists elements finite order moreover clifford plus gates applications gates circuit considered small cost compared leads measure complexity word problem consider note find universal gate sets ones topological generators form finite group together extra element take involution plus optimal respect covering small time able navigate efficiently gates key feature needed construction corresponding unit quaternion group act transitively vertices corresponding regular tree prime power extra miracle needed group act transitively edges extra requirement finitely many see section list case finite group naturally subgroup symmetries platonic solid pauli plus cube pauli matrices minimal clifford plus octahedron subgroup clifford group note generate finite index subgroup usual clifford plus group latter redundancies circuits exactly synthesizable elements unique choices hurwitz group plus tetrahedron clifford plus octahedron clifford group klein icosahedral group plus icosahedron sets enjoy relative optimal distribution navigation properties describe next difference number point view count example best presumably optimal absolute set let number circuits elements words form except possibly ends clearly want circuits represent distinct elements fact want circuits length realize distinct element elements often referred exactly synthesized gates requirement equivalent subgroup generated isomorphic case next want elements almost cover optimally almost balls respect metric volume log contain least one points order cover almost balls clearly must large almost covering essentially best possible almost circuit essentially shortest possible approximating enjoy optimal almost covering property see section see cover balls smallest size rare balls volume free exactly synthesized elements every ball volume least contain one points whether exist gate sets big hole feature interesting open problem final requirement find short circuits whose existence ensured discussion efficiently task given ball find exists circuit gates length lies problem clearly algorithm introduced executed sets leads assumption one factor integers quickly heuristic algorithm whose center diagonal matrix resolves task poly log steps particular finds given one circuit length best approximates hand diophantine problem see heart algorithm diagonal replaced general thus finding shortest circuit approximating general apparently genuinely hard complexity problem preclude efficient algorithm gives good approximation shortest circuit factoring general product different diagonal subgroups see section one produce circuits three times longer optimally short circuit removing factor remains basic open problem concerning golden gates diophantine problem underlies analysis golden gates strong approximation sums four squares simplicity restrict odd see sections let set integer solutions let unit sphere round metric normalized volume section without efficient factorization assumption one find efficiently circuit whose length times longer optimal one see show points almost cover optimally precisely number divisors ball centered volume among ingredients proof ramanujan conjectures theorems cases hand computational complexity problem associated strong approximation see section task given find task clearly theorem shows least radomized reduction hand special form adaption given section resolves task efficiently poly log steps algorithm assumes one efficient algorithm factor running time relies heuristics see section note analogous problem optimal topological generator golden solution rotation one could start two reflections whose composition rotation covering volume words length sup interval shown lim equality iff golden ratio moreover continued fraction algorithm allows one find efficiently best approximates given end introduction brief outline paper section computational complexity results connected task established section optimal covering properties solutions generalizations number rings proven section applies results unit groups quaternions verifying advertised properties golden gate sets section devoted construction gate sets finally section examine semigroups asymmetric random walks associated gates use examples section construct ramanujan cayley digraphs acknowledgement authors thank regev illuminating discussions integer programming sums squares complexity sums two squares algorithm navigating using golden gates based solving simplest quadratic diophantine inequalities begin setting integers pekk sum two squares iff odd prime factor odd congruent mod task solve efficiently efficiently mean polynomial time input case input specifying requires log bits task find satisfying log steps fixed generally denote height rational number max log log measures number bits discussed introduction assume throughout factoring done efficiently following due schoof theorem task efficient resolution proof first factor pekk note log odd mod solve gives log algorithm solve one could also proceed instead various random running time algorithms run even faster work well practice iyj hand find solution taking iyj pej even odd modify task adding inequality complexity problem changes dramatically task iii given find solution task iii plainly say one recognize solution efficiently one presented theorem task iii randomized reduction says randomized reduction see task iii efficient solution upshot unlike factoring task iii genuinely hard proof diophantine approximation condition task iii replaced iii find arg show iii resolved efficiently long known subsum knapsack called problem task given log seek steps algorithm let maxj seek primes ring arg near chosen according number primes asymptotic note bit content task log uniformly hence arg arg discuss find efficiently leading randomized reduction part arg arg arg hence iff arg let iyj solves task iii iff conjugation argz complex note log input task iii polynomial terms task iii remains find efficiently find prime sector arg choose random section probability prime thus sampling checking prime produce requisite polynomial steps way produce using randomized polynomial time procedure determined polynomial steps thus task iii reduced task iii albeit randomized reduction remark similar problem discussed asserts task given find integers randomized reduction see discussion sums four squares diophantine approximation problem directly connected navigation four squares well known every integral solutions question finding solution efficiently discussed randomized efficient algorithm choose random check prime mod last done efficiently yes schoof gives solution prime repeat another choice probability success view density primes leads randomized algorithm approximation problem project solutions onto sending comes round metric relative measurements made example ball centered radius task given find solution remark terms bit content height take rational max task clearly use theorem show theorem task randomized reduction proof interest four squares proof inductive formulate task sum theorem asserts claimed theorem true true explain case general case similar let input task iii show resolve efficiently assuming task done efficiently choose let hence equality iff particular closest solutions correspond exactly solutions seek solution closest determining closest small enough positive one checks determination efficient hence follows efficient solution task yields one theorem limits one efficiently general good news special cases task done ross selinger give algorithm see also theorem heuristic efficient algorithm solve task discussion precise meaning heuristic clarified follows important extra feature algorithm implemented runs terminates quickly similar algorithm devised implemented setting namely navigating related ramanujan graphs key property far special solving linear constraint depending hence equivalent way tasks decoupled proceed first finding candidate solution solving problem finding integer lattice points lie convex miniscus region defined inequalities listing solutions one time done polynomial cost algorithm due lenstra applies integer convex programming problems fixed dimension see key input minkowski reduction bases lattices input complexity task examine solutions geometric ordering randomly check solution running efficient solution task arrive solution resolved task fixed number steps cover solutions found output task solution case stumbling block algorithm terminating efficiently polynomial number solutions inspections produce solutions particular would happen fact solutions among many solutions heuristic aspect algorithm last scenario happen density numbers sums two squares log one expects solution probability thus probability hitting stumbling block tries small possibly never arises completes analysis meaning theorem remark theorem elsewhere assumed one factor efficiently without appealing factoring algorithm seek approximation require prime density special solutions bit smaller rather log find solutions efficiently sums squares number rings coordinates golden gates sets lie ring certain number fields leads examination tasks discussed sections replaced ring integers number fields unique factorization domains fact even euclidean domains allows extend results previous sections rings explicate need later task solve fixed given factoring norm primes applying obtain efficient factorization primes thus left dealing case totally positive prime using euclidean algorithm compute gcd situation needing find mod apply yields efficient solution task address task form theorem solve totally positive galois conjugate decouples points form rank lattice inequalities defining set lie convex compact set lenstra algorithm allows find points efficiently setting instance given task efficient solution rest analysis shown theorem valid strong approximation integer points spheres spectral gap strong approximation interested well points satisfying cover powerful method address makes use points corresponding orbits defining hecke operators eigenvalues may estimated optimally using ramanujan conjectures see formulate covering estimate terms spectral gap general compact topological group clarifies roles ingredients apply let haar measure normalized probability measure let finite subset cardinality interested right cover say pair covers interest allow everything including change say almost covering clear almost covering must chosen random necessary sufficient almost covering let let probability measure given point mass defines right convolution operator clearly orthogonal space kts invariant let estimate covering properties terms spectral norm particular proposition almost covering proof indicator function set let hence well known property coupon collector problem bracketed expression hence gives claimed inequality continuous group choose satisfy hence proposition applications size namely case iff matching hence cases essentially optimal almost covering property conclude actually covering need assume shape specifically take ball case interest let metric finite discrete metric let ball centered volume ball centered volume covering iff every ball volume contains corollary ball center radius volume covering ball centered radius proof thats meets hence lies measure set least hand proposition contradicts assumption remarks continuous group case small possibly corollary yields every ball volume contains point cases covering volume bound might far truth natural cases see sharp finite balls singletons casesif upper bound proposition gives lower bound almost covering asserts covering proposition asserts variants proposition symmetric spaces see cayley graphs formulated main variance inequality terms spectral norm alone important cases one exploit explicit version normal hence diagonalizable equal eigenvalue corresponding follows invoking form one allow exceptions long many one also remove high frequency continuous setting since small analysis carried different setting first apply proposition question covering properties arguments gaussian primes discussed section consider set primes consider distinct prime number theorem log arguments elements circle group apply proposition corollary give weight log hence log point weights assuming generalized riemann hypothesis hecke see log hence log example high frequencies estimated easily find log log hence log log almost covering almost intervals log length except extra log factor sharp direct bearing randomized reduction section namely find requisite sector choose random angle therein according prime much smaller sector area log check efficiently integer lattice points sector see prime course still randomized reduction since almost covering length optimally small grh natural question small covering length proposition coupled grh gives essentially log shows covering length log turns upper bound optimal let two close distinct primitive elements arguments tan hence fixed small norm tan interval length free primitive hence certainly free log prime thus upper bounds almost covering length log covering length sharp log factors quaternions ramanujan let denote hamilton quaternions projection morphism onto quaternions norm moreover given isomorphism metrics isometric way solutions denote give points apply results section study covering properties balls assume odd jacobi showed see convolution operator take ramanujan conjectures deligne theorem imply see also discussion hence almost covering two special cases interest prime suffices matches case fixed prime logp suffices almost cover last case interest covering properties golden gates log factor optimal far covering properties apply together corollary get suffices cover balls volume interesting question determine covering volume least exponent elementary argument using repulsion projections points onto see shows balls volume free points hence exponent covering volume lies interval probably equal one establish exponent using kloosterman circle method see approach kloosterman sums estimates associated play central role assuming natural conjecture cancellations sums involving one establish volume covering exponent points badly approximable leads rich metric diophantine approximation theory much general contexts see turn questions integers number field let totally real number field degree denote galois embeddings identity definite quadratic form sum four squares use hilbert modular form theory study distribution solutions let denote group units iff thus properties interest except special see class number form one exact number solutions simple divisor function however asymptotic behavior principal ideal generated prime factors finitely many ramified primes given divisor sum positive constant related finite group auto norm ideal note lies identified hence via diagonal embedding gives point analysis beginning section ramanujan conjectures valid well show applying proposition deduce almost covers essentially optimal covering two two extreme cases interest ball radius require little bigger order almost balls radius contain point case interest fixate first factor take ball radius long goes infinity required almost covers case need little larger applied special cases suffices analysis optimal covering properties golden gate circuits purpose navigation exact circuit length need exploit special quadratic forms class number one arise norm forms quaternion algebras give different treatment point view local global arithmetic special quaternion algebras also brings important special finite subgroups used define gates let real completion prime ideal let completion let totally definite quaternion algebra standard hamilton quaternion algebra primes unramified ramified primes even number one include count choose order usually take maximal basic properties definitions see let denote corresponding adele ring order unramified interested group outside via identification projects onto lattice see also using embeddings get diagonal embedding quotient space regular tree acts isometrically let space dgp functions hecke operators acting second variable defined preserves also orthocomplement constant function self adjoint eigenvalues satisfy one way derive use automorphic representations eigenfunctions diagonalization hecke operators primes embedded vector non irreducible representation occuring right regular representation ramanujan conjectures theorems context thanks correspondence deligne theorem see assert ranging places tempered particular applies spherical representation turn implies eigenvalue satisfies connect circuit lengths elements impose strong condition namely action transitive happen outside finite set primes class number order needs definite quaternion algebra setting happens finitely many enumerated transitive case may identified functions invariant identity coset explicitly unique extension action becomes according kind operator considered sharp spectral bound finally transitive setting relate distance moved tree circuit word length generators let elements take immediate neighbors hence immediate neighbors unique repeat repeat arrive hence since every step choices determined unique expression form thus distance moved tree corresponds word length generators hecke orbit corresponds circuit given length moreover given want express form done efficiently navigating tree described important feature exploited navigation algorithms case acts set simply transitively invariant hence shows cayley graph respect generators infinite order distance tree corresponding word circuit length symmetric set generators case sets noted arise finite number see section examples varying infinitely many golden gate sets super golden gate sets require acts transitively edges fact assume acts simply transitively among involution case set taken note asserts exactly representation product gives rise gate sets plus finitely many since set lie finite list set orders hence unit group order limited limits primes group act transitively conclude finitely many possibilities super golden gate sets number interesting examples happens described next section end section describing relation word length elements terms generators strong approximation explain case hamilton quaternions case highlighted basic example calling let order group invertible elements spheres correspond solutions spheres acted left right multiplication odd prime multiplying right solutions exactly one coordinate odd use make first coordinate also ensure positive gives solutions see exactly one coordinates even use arrange coordinate first one act choose sign way solutions obtained become see unramified odd prime group generated elements realized subgroup generators take identity coset neighbors acts simply transitively moreover cayley graph word length generators yields unique right multiplication primitive namely gcd furthermore every primitive achieved way identifies distance dxp word length generators splitting isomorphism yields subgroup also denoted generated elements question well words length cover equivalent question covering problem discussed beginning section follows circuits length almost cover optimally precisely number circuits length fixed long almost balls covered finally far navigating generators problem reduces one strong approximation namely task section task needs resolved poly steps determine circuit length separately question given decouples two steps firstly finding according theorem done efficiently least special amounts diagonal found determined none exists express circuit length generators viewing unique generator move one step closer repeating times gives efficient factorization summarize given diagonal assuming factor efficiently heuristic discussion theorem find efficiently circuit least word length generators lies without efficient factorization algorithm settle finding special solutions strong approximation problem described remark otherwise proceed secures circuit lies whose word length times longer optimally short circuit allows navigate diagonal optimally efficiently far optimal navigation general via strong approximation problem least form task apparently hard navigate efficiently least two coordinate equal particular elements form pauli group diagonal element expressed product three efficiently hence navigate efficiently circuit whose lengthy longer shortest circuit generic efficient navigation super golden gates using carried similarly completes theoretical diophantine algorithmic analysis concerning golden gates announced introduction end section two explicit examples take generators yield golden gate set called yield four involutions given golden gate set action given transitive leads first example super gate turn super golden gates set composed finite group involution lie group prime ideal integers totally definite quaternion algebra totally real number field require acts simply transitively neighbors origin involution takes origin one neighbors since involution inverts edge denote group generated group generated next proposition follows theory assumptions proposition acts simply transitively directed edges dxp orig dxp particular elements free product set namely acts simply transitively vertices thus free product copies one navigate follows origin unique pointing towards since rotations around origin dxp orig dxp term dxp orig dxp orig continuing manner one finds orig hence examples examples sets follow explicate definite quaternion algebra totally real field order well prime allow denominators class number refer corresponding entries table lipschitz quaternions pauli matrices simplest example occurs hamilton quaternions lipschitz order used construct golden gates previous section one takes explicit isomorphism given unit group corresponds pauli matrices mapped indeed act simply transitively neighbors one verify multiplying right computing iwasawa decomposition result element involution inverts edge using corresponds hurwitz quaternions tetrahedral gates consider maximal order hurwitz quaternions keeping corresponds entry tab unit group platonic tetrahedral group isomorphic corresponds splitting used acts simply transitively neighbors origin involution one take maps examples follow write explicit action tree examples splitting chosen fixes vertex okp invert edge incident octahedral gates take hamilton quaternions maximal order given entry tab see also unit group modulo scalars platonic octahedral group corresponds clifford group unique continuous isomorphism given sending square root chosen maps uniformizer splitting given adding involution corresponds gives set full group subgroups give obtain gates correspond involution ramified prime one splitting given correspond gate set obtained hybrid example taking subgroup octahedral group one scale generators lie lipschitz quaternions generate infinite group projects onto copy together involution gives gate set icosahedral gates move golden field maximal order given ring icosians mod mod mod tab forms respect quadratic form copy lattice unit group modulo scalars platonic icosahedral group corresponds one splitting used involution one take gives generated group full group icosahedral taking one using gives another set gates acting tree corresponding involution inert taking completion residue field size four splitting given cyclic group involution give gate set nonexamples ramified inert primes corresponding groups act tree respectively involutions invert edge origin groups act transitively neighbors remark focus throughout optimal covering exponent navigation one relaxes class number assumption consists single prime one loses optimal features however gate sets arise general groups coming definite quaternion algebras still reasonably good covering exponents navigation properties studied operators section study asymmetric hecke operators tree whose spectral properties coincides random walk nbrw ramanujan graph explored corresponds words circuits free semigroups generated golden gate sets ramanujan semigroups set edge flipped section denote sector descended namely vertices shortest path begins let observe semigroup generated assuming leading away origin map gives correspondence edges lead level particular free semigroup spectrum notation section particularly nice spec compared thm asserts symmetric set size max spec latter bound corresponds spectral radius random walk cayley graph free group bound spectral radius random walk free semigroup furthermore since precisely tsr obtain every spec one note however normal enough determine spectral norm fact turns however well obtained considering action tree applying strong approximation ramanujan conjectures describe case corresponds standard hamilton quaternions simplicity since acts simply transitively edges iwahori subgroup gqp namely stabgp via isomorphism representation space iwahorihecke algebra algebra compactly supported functions edges convolution acts element acts algebra ramanujan conjectures imply every irreducible representation appears tempered matrix coefficients vectors edges operator corresponds nbrw thus spectrum contained nbrw regular tree furthermore every tempered iwahorispherical unitary irreducible representation either twist steinberg representation case dim principal series representation induced norm case dim appropriate orthonormal basis follows remark decomposition graph setting key ingredient ramanujan digraphs sets used construct explicit ramanujan graphs finite graphs whose nontrivial spectrum contained within tree namely every eigenvalue adjacency operator either trivial satisfies similar manner set defined used construct ramanujan digraphs say digraph ramanujan every spec adjg satisfies words nontrivial spectrum contained within cayley digraph free semigroup generators ramanujan digraphs see one examples section let cayley graph hck respect naturally identified directed line graph tree namely vertices correspond directed edges edges steps adjacency operator cay describes nbrw tree ideal let ker gok let cayley graph respect set generators verts representation algebra ramanujan conjectures imply every nontrivial eigenvalue adj corresponds vector tempered representation giving section addition trivial eigenvalue arises trivial representation unlike continuous case also obtain eigenvalue whenever quadratic finally note groups identified concretely denoting wedderburn theorem gfq gfq either according legendre symbol example taking see section one obtains gives similarly spectrum ramanujan digraph obtained shown figure figure adjacency spectrum cayley graph references manindra agrawal neeraj kayal nitin saxena primes annals mathematics pages andreas blass alex bocharov yuri gurevich optimal circuits axial rotations journal mathematical physics sergey bravyi alexei kitaev universal quantum computation ideal clifford gates noisy ancillas phys rev browning vinay kumaraswamy steiner twisted linnik implies optimal covering exponent arxiv preprint alex bocharov krysta svore quantum circuits phys rev nov jean bourgain peter sarnak rudnick local statistics lattice points sphere modern trends constructive function theory contemp math cadilhac variant factoring http henri cohen course computational algebraic number theory volume graduate texts mathematics berlin conway sloane 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optimal approximation single qubit unitaries clifford circuits using constant number ancillary qubits physical review letters vadym kliuchnikov dmitri maslov michele mosca fast efficient exact synthesis unitaries generated clifford gates quantum information computation kubilius problem analytic theory numbers viliniaus valst univ mokslo dardai chem moksly ser markus kirschmer john voight algorithmic enumeration ideal classes quaternion orders siam journal computing lagrange nouv acad berlin oeuvres hendrik lenstra integer programming fixed number variables mathematics operations research eyal lubetzky alex lubotzky ori parzanchevski random walks ramanujan complexes digraphs arxiv preprint algorithmic theory numbers graphs convexity volume regional conference series applied mathematics siam philadelphia eyal lubetzky yuval peres cutoff ramanujan graphs geometric functional analysis lubotzky phillips sarnak hecke operators distributing points sphere comm pure appl lubotzky phillips sarnak hecke operators distributing points comm pure appl lubotzky phillips sarnak ramanujan graphs combinatorica michael nielsen isaac chuang quantum computation quantum information cambridge university press new york usa edition christophe petit kristin lauter quisquater full cryptanalysis lps morgenstern hash functions international conference security cryptography networks pages springer neil ross optimal approximation quantum information computation michael rabin jeffery shallit randomized algorithms number theory comm pure appl suppl frontiers mathematical sciences new york neil ross peter selinger optimal approximation zrotations quantum information computation igor rivin naser talebizadeh sardari preparation peter sarnak number points certain curves uncertainty principle number theory pages naser talebizadeh sardari diameter ramanujan graphs random cayley graphs numerics arxiv preprint naser talebizadeh sardari optimal strong approximation quadratic forms arxiv preprint sarnak appendix letter optimal lifting integral points http sarnak letter aaronson pollington theorem golden gates http naser talebizadeh sardari complexity strong approximation sphere arxiv preprint satake spherical functions ramanujan conjecture proc sympos pure math volume pages schoof elliptic curves finite fields computation square roots mod math serre trees york translated french john stillwell rainer representation integral quadratic forms survey contemporary mathematics sudan problem http jacques tits quaternions leech lattice sporadic group halljanko journal algebra des quaternions volume lecture notes mathematics springer berlin
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oct deep motif dashboard visualizing understanding genomic sequences using deep neural networks jack lanchantin ritambhara singh beilun wang yanjun department computer science university virginia charlottesville abstract deep neural network dnn models recently obtained prediction accuracy transcription factor binding tfbs site classification task however remains unclear approaches identify meaningful dna sequence signals give insights tfs bind certain locations paper propose toolkit called deep motif dashboard demo dashboard provides suite visualization strategies extract motifs sequence patterns deep neural network models tfbs classification demonstrate visualize understand three important dnn models convolutional recurrent networks first visualization method finding test sequence saliency map uses derivatives describe importance nucleotide making final prediction second considering recurrent models make predictions temporal manner one end tfbs sequence introduce temporal output scores indicating prediction score model time sequential input lastly visualization strategy finds optimal input sequence given tfbs positive class via stochastic gradient optimization experimental results indicate architecture performs best among three architectures visualization techniques indicate makes predictions modeling motifs well dependencies among introduction recent years explosion deep learning models lead groundbreaking results many fields computer vision natural language processing computational biology however although models proven accurate widely viewed black boxes due complexity making hard understand particularly unfavorable biomedical domain understanding model predictions extremely important doctors researchers trying use model aiming open black box present deep motif demo dashboard understand inner workings deep neural network models genomic sequence classification task introducing suite different neural models visualization strategies see ones perform best understand make understanding genetic sequences one fundamental tasks health advancements due high correlation genes diseases drugs important problem within genetic sequence understanding related transcription factors tfs regulatory proteins bind dna different binds specific transcription factor binding sites tfbss genome regulate cell machinery given input dna sequence classifying whether binding site particular core task bioinformatics dashboard normally refers user interface gives current summary usually graphic form key information relating performance implemented model torch made available task follow two step approach first given particular interest dataset containing samples positive negative tfbs sequences construct three deep learning architectures classify sequences section introduces three different dnn structures use convolutional neural network cnn recurrent neural network rnn neural network trained models predict binding sites second step approach understand models perform way explained section introducing three different visualization strategies interpreting models measuring nucleotide importance saliency maps measuring critical sequence positions classifier using temporal output scores generating motif patterns class optimization test evaluate models visualization strategies large scale benchmark tfbs dataset section provides experimental results understanding visualizing three dnn architectures find outperforms models visualizations observe tends focus predictions traditional motifs well modeling long range dependencies among motifs deep neural models tfbs classification tfbs classification chromatin immunoprecipitation technologies databases encode made binding site locations available hundreds different tfs despite advancements two major drawbacks experiments slow expensive although experiments find binding site locations find patterns common across positive binding sites give insight tfs bind locations thus need large scale computational methods make accurate binding site classifications also identify understand patterns influence binding site locations order computationally predict tfbss dna sequence researchers initially used consensus sequences position weight matrices match test sequence simple neural network classifiers proposed differentiate positive negative binding sites show significant improvements weight matrix matching methods later svm techniques outperformed generative methods using features string kernel based svm systems limited expensive computational cost proportional number training testing sequences recently convolutional neural network models shown results tfbs task scalable large number genomic sequences remains unclear neural architectures work best deep neural networks tfbss find neural models work best tfbs classification task examine several different types models inspired success across different fields explore variations two popular deep learning architectures convolutional neural networks cnns recurrent neural networks rnns cnns dominated field computer vision recent years obtaining results many tasks due ability automatically extract features hand rnns emerged one powerful models sequential data tasks natural language processing due ability learn long range dependencies specifically tfbs prediction task explore three distinct architectures cnn rnn combination two figure shows overview models deep framework body three architectures use differ implemented model follows similar framework use easily compare contrast results use raw nucleotide characters inputs character converted encoding binary vector matching character entry rest encoding matrix used input convolutional recurrent module outputs vector fixed dimension output vector model linearly fed softmax function last layer learns mapping hidden space output class label space final output probability indicating whether input positive negative binding site binary classification task parameters network trained minimizing negative training set minimization loss function obtained via stochastic gradient algorithm adam size sequences use dropout regularization method model tfbs tfbs linear softmax tfbs linear softmax average concatenate linear softmax maxpool average concatenate lstm acttgcag acttgcag cnn lstm acttgcag rnn figure model architectures model input encoded matrix raw nucleotide inputs output softmax classifier make binary prediction architectures differ middle module convolutional recurrent convolutional neural network cnn genomic sequences believed regulatory mechanisms transcription factor binding influenced local sequential patterns known motifs motifs viewed temporal equivalent spatial patterns images eyes face cnns able automatically learn achieve art results computer vision tasks result temporal convolutional neural network fitting model automatically extract motifs temporal convolution filter kernel size takes input data matrix size nin length input layer size nin outputs matrix size nout nout output layer size specifically convolution nin trainable parameters convolution filter function enforcing nonlinearity use rectified linear units relu nonlinearity relu max convolution nonlinearity cnns typically use maxpooling dimension reduction technique provide translation invariance extract higher level features wider range input sequence temporal maxpooling matrix pooling size results output matrix formally maxpool max cnn implementation involves progression convolution nonlinearity maxpooling represented one convolutional layer network test layer deep cnns final layer involves maxpool across entire temporal domain vector fed softmax classifier figure shows cnn model two convolutional layers input encoded matrix convolved several filters shown fed relu nonlinearity produce matrix convolution activations perform maxpool activation matrix output first maxpool fed another convolution relu maxpooled across entire length resulting vector vector transposed fed linear softmax layer classification recurrent neural network rnn designed handle sequential data recurrent neural networks rnns become main neural model tasks natural language understanding key advantage rnns cnns able find long range patterns data highly dependent ordering sequence prediction task given input matrix size nin rnn produces matrix size rnn embedding size timestep rnn takes input column vector rnin previous hidden state vector produces next hidden state applying following recursive operation wxt trainable parameters model nonlinearity due recursive nature rnns model full conditional distribution sequential data find dependencies time position sequence timestep imaginary time coordinate running certain direction handle vanishing gradients problem training basic rnns long sequences hochreiter schmidhuber proposed rnn variant called long memory lstm network simplicity refer lstms rnns paper handle long term dependencies using gating functions gates control information written read forgotten specifically lstm cells take inputs produce tanh tanh tanh sigmoid hyperbolic tangent multiplication functions respectively input forget output gates respectively rnns produce output vector timestep input sequence order use classification task take mean vectors use mean vector hmean input softmax layer since innate direction genomic sequences use lstm rnn model lstm input sequence gets fed two lstm networks one direction output vectors direction get concatenated together temporal direction fed linear classifier figure shows rnn model input encoded matrix fed lstm forward backward direction produce matrix column vectors representing lstm output embedding timestep vectors averaged create one vector direction representing lstm output forward backward output vectors concatenated fed softmax classification network considering convolutional networks designed extract motifs recurrent networks designed extract temporal features implement combination two order find temporal patterns motifs given input matrix output cnn column vector gets fed rnn one time way encoded vectors get input regular rnn model resulting output rnn lstm embedding size averaged across temporal domain way regular rnn fed softmax classifier figure shows model input encoded matrix fed one layer convolution produce convolution activation matrix matrix input lstm done regular rnn model original matrix output lstm averaged concatenated fed softmax similar rnn visualizing understanding deep models previous section explained deep models use tfbs classification task evaluate models perform best making accurate predictions important biomedical tasks equally important understand models make predictions accurate uninterpretable models often slow emerge practice due inability understand predictions making biomedical domain experts reluctant use consequently aim obtain better understanding certain models work better others investigate make predictions introducing several visualization techniques proposed demo dashboard allows visualize understand dnns three different ways saliency maps temporal output scores class optimizations saliency maps certain dna sequence model classification logical question may parts sequence influential classification seek visualize influence position nucleotide prediction approach similar methods used images simonyan baehrens given sequence length class dnn model provides score function rank nucleotides based influence score since highly function deep neural nets hard directly see influence nucleotide mathematically around point approximated linear function computing taylor expansion derivative respect sequence variable point saliency map derivative simply one step backpropagation dnn model therefore easy compute pointwise multiplication saliency map encoded sequence get derivative values actual nucleotide characters sequence see influence character position output score finally take magnitude resulting derivative vector visualize important character regardless derivative direction call resulting vector saliency map tells nucleotides need changed least order affect class score see equation saliency map simply weighted sum input nucleotides weight indicates influence nucleotide position output score temporal output scores since dna sequential read certain direction insightful visualize output scores timestep position sequence call temporal output scores assume imaginary time direction running left right given sequence position sequence timestep imagined time coordinate words check rnn prediction scores vary input rnn input series constructed using subsequences input running along imaginary time coordinate subsequences start first nucleotide position ends entire sequence way see exactly sequence recurrent model changes decision negative positive vice versa since recurrent models also use technique reverse sequence cnns process entire sequence thus view output temporal sequence use visualization rnn class optimization previous two visualization methods listed representative specific testing sample introduce approach extract visualization dnn model attempt find best sequence maximizes probability positive tfbs call class optimization formally optimize following equation probability score input sequence matrix case positive tfbs computed softmax equation trained dnn model specific arg max regularization parameter find locally optimal stochastic gradient descent optimization respect input sequence optimization model weights remain unchanged similar methods used simonyan optimize toward specific image class visualization method depicts notion positive tfbs class particular specific test sequence automatic motif extraction dashboard three proposed visualization techniques allow manually inspect models make predictions order automatically find patterns techniques also propose methods extract motifs consensus subsequences represent positive binding sites extract motifs three visualization methods following ways positive test sequence thus total dataset extract motif saliency map selecting contiguous subsequence achieves highest sum contiguous saliency map values positive test sequence extract motif temporal output scores selecting subsequence shows strongest score change negative positive output score different directly use sequence motif connecting previous studies neural networks produced results several important benchmark tasks related genomic sequence classification making good candidate use however models work well poorly understood recent works attempted uncover properties models work done understanding image classifications using convolutional neural networks zeiler fergus used deconvolution approach map hidden layer representations back input space specific example showing features image important classification simonyan explored similar approach using taylor expansion linearly approximate network find input features relevant also tried optimizing image classes many similar techniques later followed understand convolutional models importantly researchers found cnns able extract layers feature maps may indicate cnns successfully used genomic sequence predictions believed triggered motifs tasks fewer visualization studies dnns karpathy explored interpretability rnns language modeling found exist interpretable neurons able focus certain language structure quotes visualized rnns achieve compositionality natural language sentiment analysis visualizing rnn embedding vectors well measuring influence input words classification studies show examples validated understanding natural language linguistics contrarily interested understanding dna linguistics given dnns opposite direction karpathy main difference work previous works images natural language instead trying understand dnns given human understanding human perception tasks attempt uncover critical signals dna sequences given understanding dnns tfbs prediction alipanahi first implement visualization method dnn model visualize cnn model extracting motifs based input subsequence corresponding strongest activation location convolutional filter call convolution activation since one convolutional layer trivial map activations back method work well deeper models attempted technique models found approach using saliency maps outperforms finding motif patterns details section quang xie use visualization method model noncoding variant prediction experiments results experimental setup dataset order evaluate dnn models visualizations train test cell encode datasets used alipanahi dataset average training sequences even split sequence consists characters every dataset testing sequences even split positive sequences extracted genome centered reported peak negative sequences generated table variations dnn model hyperparameters conv conv conv filter conv pool model layers size nout sizes size small rnn medium rnn large rnn small cnn medium cnn large cnn small medium large lstm layers lstm size table mean auc scores tfbs classification task model deepbind cnn small rnn med rnn large rnn small cnn med cnn large cnn small med large mean auc median auc stdev table auc pairwise model rnn meme cnn meme meme cnn rnn rnn cnn shuffle positive sequences due separate data train separate model individual dataset variations dnn models implement several variations dnn architecture varying hyperparameters table shows different hyperparameters architecture trained many different hyperparameters architecture show best performing model type surrounded larger smaller version show underfitting overfitting baselines use sum results alipanahi one prediction performance baseline results applying top positive training sequences deriving five pwms scoring test sequences using sum scores using five pwms also compare cnn model proposed alipanahi evaluate motif extraction compare convolution activation method used alipanahi quang xie map strongest first layer convolution filter activation back input sequence find influential subsequence tfbs prediction performance dnn models table shows mean area roc curve auc scores tested models table expected cnn models outperform standard rnn models validates hypothesis positive binding sites mainly triggered local patterns motifs cnns easily find interestingly achieves best performance among three deep architectures check statistical significance comparisons apply pairwise using auc scores report two tailed table apply best performing based auc models model type deep models significantly better meme baseline cnn significantly better rnn significantly better cnn order understand performs best turn dashboard visualizations figure demo dashboard dashboard examples mafk nfyb positive tfbs sequences top section dashboard contains class optimization pertain specific test sequence rather class general middle section contains saliency maps specific positive test sequence bottom section contains temporal output scores positive test sequence used saliency map top contains known jaspar motifs highlighted pink boxes test sequences contain motifs table jaspar motif matches demo dashboard baseline motif finding methods using tomtom saliency map conv activations temporal output class optimization cnn rnn understanding dnns using demo dashboard evaluate dashboard visualization methods first manually inspect dashboard visualizations look interpretable signals figure shows examples demo dashboard three different tfs positive tfbs sequences apply visualizations best performing models three dnn architectures dashboard snapshot specific contains jaspar motifs gold standard motifs generated biomedical researchers positive tfbs sequence architecture given interest positive tfbs test sequence interest jaspar motifs test sequences highlighted using pink box saliency map dnn model test sequence forward backward temporal output scores recurrent architectures test sequence saliency maps red position influential prediction temporal outputs blue indicates negative tfbs prediction red indicates positive saliency map temporal output visualizations positive test sequence shown twice numbers next model names saliency map section indicate score outputs dnn model specified test sequence saliency maps middle section dashboard visual inspection see saliency maps cnns tend focus short contiguous subsequences predicting positive bindings words cnns clearly model motifs influential prediction saliency maps rnns tend spread across entire sequence indicating focus nucleotides together infer relationships among strong saliency map values around motifs also see nucleotides away motifs influential model prediction example model confident tfbs prediction prediction also influenced nucleotides outside motif mafk saliency maps see rnn focus wide range nucleotides make predictions rnn even focus known jaspar motif make high confidence prediction temporal output scores bottom section dashboard sequences tested positions trigger model switch negative tfbs prediction positive near jaspar motifs observe clear differences forward backward temporal output patterns certain cases interesting look temporal output scores saliency maps together important case study examples nfyb example cnn rnn perform poorly makes correct prediction observe able switch classification negative positive rnn never understand may happened see saliency maps focuses two distinct regions one flips classification negative positive however rnn focus either areas may reason never able classify positive sequence fact cnn able classify positive sequence focuses regions saliency map may indicate temporal dependencies regions influence binding addition fact clear jaspar motif sequence may show traditional motif approach always best way model tfbss class optimization top section dashboard class optimization cnn model generates concise representations often resemble known motifs particular recurrent models tfbs positive optimizations less clear though aspects stand like followed notice certain dnn models class optimized sequences optimize reverse complement motif nfyb cnn optimization class optimizations useful getting general idea triggers positive tfbs certain automatic motif extraction dashboard order evaluate dnn capability automatically extract motifs compare found motifs method introduced section corresponding jaspar motif interest comparison using tomtom tool searches query motif given motif database reverse complements returns significant matches ranked indicating similarity table summarizes motif matching results comparing motifs known motifs jaspar database limited comparison datasets tfs jaspar motifs compare four visualization approaches saliency map convolution activation temporal output scores class optimizations first three techniques sequence specific therefore report average number motif matches positive sequences averaged across datasets last technique particular tfbs positive class see table across multiple visualization techniques cnn finds motifs best followed rnn however since cnns perform worse auc scores hypothesize demonstrates also important model sequential interactions among motifs combination cnn acts like motif finder rnn finds dependencies among motifs analysis shows visualizing dnn classifications lead better understanding dnns tfbss conclusions future work deep neural networks dnns shown accurate models tfbs classification however dnn models hard interpret thus adaptation practice slow work propose deep motif demo dashboard explore three different dnn architectures tfbs prediction introduce three visualization methods shed light models work although visualization methods still require human practitioner examine dashboard start understand models hope work invoke studies visualizing understanding dnn based genomic sequences analysis furthermore dnn models recently shown provide excellent results epigenomic analysis plan extend demo dashboard related applications references dashboard definiton http accessed babak alipanahi andrew delong matthew weirauch brendan frey predicting sequence specificities proteins deep learning nature publishing group sebastian bach alexander binder montavon frederick klauschen wojciech samek explanations classifier decisions relevance propagation volume page david baehrens timon schroeter stefan harmeling motoaki kawanabe katja hansen explain individual classification decisions volume pages encode project consortium integrated encyclopedia dna elements human genome volume pages nature publishing group mahmoud ghandi dongwon lee morteza michael beer enhanced regulatory sequence prediction using gapped features shobhit gupta john stamatoyannopoulos timothy bailey william noble quantifying similarity motifs volume page biomed central ltd sepp hochreiter schmidhuber long memory volume pages mit press paul horton minoru kanehisa assessment neural network statistical approaches prediction coli promoter sites volume pages oxford univ press andrej karpathy justin johnson visualizing understanding recurrent networks david kelley jasper snoek john rinn basset learning regulatory code accessible genome deep convolutional neural networks cold spring harbor lab diederik kingma jimmy adam method stochastic optimization alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks advances neural information processing systems pages jack lanchantin ritambhara singh zeming lin yanjun deep motif visualizing genomic sequence classifications jiwei xinlei chen eduard hovy dan jurafsky visualizing understanding neural models nlp philip machanick timothy bailey motif analysis large dna datasets volume pages oxford univ press aravindh mahendran andrea vedaldi visualizing deep convolutional neural networks using natural pages springer anthony mathelier oriol fornes david arenillas chen denay jessica lee wenqiang shi casper shyr tan rebecca jaspar major expansion update database transcription factor binding profiles page oxford univ press daniel quang xiaohui xie danq hybrid convolutional recurrent deep neural network quantifying function dna sequences page cold spring harbor labs journals manu setty christina leslie seqgl identifies binding signals regulatory element maps karen simonyan andrea vedaldi andrew zisserman deep inside convolutional networks visualising image classification models saliency maps ritambhara singh jack lanchantin gabriel robins yanjun deepchrome predicting gene expression histone modifications volume pages nitish srivastava geoffrey hinton alex krizhevsky ilya sutskever ruslan salakhutdinov dropout simple way prevent neural networks overfitting volume pages gary stormo dna binding sites representation discovery volume pages oxford univ press ilya sutskever oriol vinyals quoc sequence sequence learning neural networks advances neural information processing systems pages matthew zeiler rob fergus visualizing understanding convolutional networks computer pages springer jian zhou olga troyanskaya predicting effects noncoding variants deep sequence model volume pages nature publishing group
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aug new extremal singly even codes lengths damyan masaaki nikolay august abstract lengths construct extremal singly even codes weight enumerators extremal singly even selfdual codes previously known exist also construct new inequivalent extremal doubly even codes covering radius meeting delsarte bound introduction binary code vector subspace denotes finite field order codes note binary parameter called length weight vector number components vector codeword minimum weight codewords called minimum weight code minimum weight called code dual code code length defined standard inner product code called code doubly faculty mathematics informatics konstantin preslavski university shumen shumen bulgaria research center pure applied mathematics graduate school information sciences tohoku university sendai japan faculty mathematics informatics konstantin preslavski university shumen shumen bulgaria even codewords weight divisible four singly even least one codeword mod known code length exists even doubly even code length exists divisible let singly even code let denote subcode consisting codewords mod shadow defined shadows codes introduced conway sloane order give largest possible minimum weight among singly even codes provide restrictions weight enumerators singly even codes largest possible minimum weights among singly even codes length given possible weight enumerators singly even codes largest possible minimum weights given fundamental problem find weight enumerators actually occur possible weight enumerators see considering shadows rains showed minimum weight code length bounded mod otherwise code meeting bound called extremal aim note construct extremal singly even codes weight enumerators extremal singly even codes previously known exist precisely construct extremal singly even codes weight enumerators see section codes constructed neighbors extremal singly even codes construct extremal singly even codes weight enumerators see section codes constructed extremal singly even codes method given also demonstrate least inequivalent extremal doubly even codes covering radius meeting delsarte bound computer calculations note done help algebra software magma computer system weight enumerators extremal singly even codes lengths possible weight enumerators extremal singly even selfdual codes shadows given integers extremal singly even codes weight enumerator known see extremal singly even codes weight enumerator known see possible weight enumerators extremal singly even codes shadows given integers extremal singly even codes weight enumerator known see extremal singly even codes weight enumerator known see extremal singly even codes weight enumerator known see extremal singly even codes circulant matrix following form successive row cyclic shift previous one let circulant matrices let code generator matrix following form denotes identity matrix order denotes transpose easy see aat codes generator matrices form called two codes equivalent one obtained permutation coordinates section give classification extremal singly even codes exhaustive search found distinct extremal singly even codes must checked equivalence complete classification done considering pairs circulant matrices satisfying condition aat sum weights first rows congruent mod sum weights greater equal since cyclic shift first rows gives equivalent code may assume without loss generality last entry first row computer search shows distinct extremal singly even codes divided inequivalent codes proposition equivalence extremal singly even codes denote codes codes first rows resp circulant matrices resp generator matrices listed table verified codes weight enumerator also listed table extremal neighbors two codes length said neighbors dim code length reached taking successive neighbors see since every code length contains vector subcodes codimension containing since dim two codes rather lying singly even code length divisible two doubly even neighbors see section construct extremal codes considering neighbors found distinct extremal singly even neighbors equivalent none codes verified codes divided inequivalent codes codes constructed save space values supports supp values weight enumerators listed http codes extremal singly even codes weight enumerators extremal singly even codes previously known exist supp list table hence following proposition extremal singly even code weight enumerator table extremal singly even codes codes table extremal singly even codes continued codes consider extremal doubly even neighbors since shadow minimum weight two doubly even neighbors extremal doubly even codes covering radius see thus six extremal doubly even codes covering radius constructed addition among codes extremal singly even codes shadow minimum weight constructions codes listed table two doubly even neighbors extremal doubly even codes covering radius verified following equivalent codes among four codes six codes codes means equivalent pair equivalent codes therefore following proposition proposition least inequivalent extremal doubly even selfdual codes covering radius meeting delsarte bound table extremal singly even neighbors codes supp order distinguish two doubly even neighbors list table supports supp codes constructed table extremal doubly even neighbors codes supp singly even codes neighbors using approach similar given section exhaustive search found distinct singly even codes computer search shows distinct singly even selfdual codes divided inequivalent codes proposition equivalence singly even codes denote codes codes first rows resp circulant matrices resp generator matrices obtained http following method constructing neighbors given let matrix whose rows codewords weight suppose vector even weight mxt subcode index neighbors vector codeword weight neighbor minimum weight vector satisfying obtain way verified unique vector satisfying two neighbors doubly even code case two neighbors automatically doubly even hence following proposition extremal singly even neighbor extremal singly even codes following method constructing singly even codes given let code length let vector odd weight let denote subcode consisting codewords orthogonal cosets shown code length section construct new extremal singly even codes length using construction extremal singly even codes obtained sections exhaustive search shows inequivalent extremal singly even codes constructed codes codes codes codes weight enumerator weight enumerator weight enumerator extremal singly even codes weight enumerator constructed first time four weight enumerators example codes weight enumerators given list table values codes vectors applying construction given found extremal singly even codes weight enumerators table extremal singly even codes codes extremal singly even codes previously known exist codes list table values weight enumerators codes vectors hence following proposition extremal singly even code weight enumerator weight enumerator remark code smallest value among known extremal singly even codes weight enumerator acknowledgment work supported jsps kakenhi grant number references bosma cannon playoust magma algebra system user language symbolic comput bouyukliev code equivalence advances coding theory cryptography ser coding theory world sci hackensack brualdi pless weight enumerators codes ieee trans inform theory chigira harada kitazume extremal codes length neighbors covering radii des codes cryptogr kim new cubic codes length preprint conway sloane new upper bound minimal distance codes ieee trans inform theory dougherty gulliver harada extremal binary codes ieee trans inform theory harada nishimura yorgova new extremal codes length math balkanica karadeniz yildiz new extremal binary codes length extensions codes franklin inst kaya new extremal binary codes lengths finite fields appl kaya yildiz pasa new extremal binary codes modified four circulant construction discrete math kaya yildiz siap new extremal binary codes quadratic circulant codes finite fields appl rains shadow bounds codes ieee trans inform theory tsai existence certain extremal codes ieee trans inform theory tsai extremal codes lengths ieee trans inform theory yankov codes automorphism order adv math commun yankov lee ivanova codes automorphism order ieee trans inform theory yankov lee ivanova codes automorphism order codes length finite fields submitted
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faq synthesizing command repairs examples aug loris antoni rishabh singh michael vaughn university microsoft research university loris risin vaughn abstract ing cli typing command user receives error message xxx uses set rules suggest possible fixes user command rule input command error message uses inputs produce possible fixed command typical fixes include adding missing flags creating missing directory changing file extensions xxx become extremely popular github already starred users forked times despite success tool also main limitation add new rule developer first needs understand syntax precise semantics xxx manually rule tool due complexity newly added rules times caused unexpected tools confusing hard use novice programmers due cryptic error messages lack documentation novice users often resort online finding corrections buggy commands hard time searching precisely posts relevant problem applying suggested solutions buggy command present tool faq uses set rules suggest possible fixes users write buggy commands trigger commonly occurring errors rules expressed language called ixit rule user buggy command corresponding error message uses inputs produce possible fixed command main contribution algorithm based lazy vsa synthesizing ixit rules examples buggy repaired commands algorithm allows users add new rules faq without manually encode present evaluation faq benchmark problems show faq able instantly synthesize rules benchmark problems real time using examples rule synthesizing repairs examples inspired success limitations xxx built faq frequently asked questions tool automatically addressing common errors clis faq also uses set rules fixing common errors differs xxx following key aspects rules encoded declarative language dsl called ixit introduction add new rules users provide examples buggy repaired commands faq automatically synthesizes desired ixit rules consistent given examples interfaces cli let users interact computing system writing sequences commands clis especially popular amongst advanced computer users use perform small routine tasks committing file repository version control installing software packages compiling source code finding searching files etc even though mode interaction getting replaced natural graphical user interfaces clis still routinely used scripting tasks unix mac even windows operating system officially provides complex interfaces products windows powershell since interactions often require complex parameters flag settings specifying desired intent users find clis challenging use moreover entering incorrect input command user deal cryptic errors hard decipher looking verbose documentation commands reasons users typically resort online finding corrections buggy commands unfortunately also problematic users need precisely search posts relate issues commands transform suggested solutions apply context sometimes users also need create new post relevant post exists found need wait hours days obtain solution problem envision faq system used easily extend system providing new examples fixes goal system learn examples obtained shell histories thousands users unsupervised manner although developer write similar rules system like xxx manually two main challenges first feasible easily add thousands rules generally contributor access xxx source code second even developers writing correct rules difficult complexity string manipulations needed perform command fixes fact xxx consists less rules little year since adding new ones fairly complex task ixit dsl encoding fix rules inspired types rules appearing xxx common command repairs requested users ixit rule first uses pattern matching unification match command error message applies fix transformation match succeeds transformations consist substring append functions strings present command error message present algorithm efficiently synthesizes ixit rules consistent given set examples using algebra vsa synthesis techniques used succinctly represent set expressions common errors recently tool developed automatically addressing common errors decided censor name tool tool found http http http consistent set examples even though existing vsa data structures represent exponential number ixit rules polynomial space space still quite large address problem introduce lazy algebra given set examples algorithm maintains lazy representation subset ixit rules consistent examples rules missing enumerated new example accounted adding ixit rule already present careful design ixit synthesis algorithm polynomial time complexity contrast existing synthesis techniques string transformations require exponential time polynomial time complexity crucial synthesis algorithm scale large number fix examples since different examples may refer different target rules propose strategy partition input examples groups examples corresponding individual rules use lazy vsa algorithm learn ixit rules partition evaluate synthesis algorithm implemented faq benchmark problems obtained xxx online faq able learn repair rules buggy commands benchmark problems examples encounter similar errors example let say skilled developer provides another triple following form javac pair class names pair accepted annotation processing explicitly requested javac using two examples faq synthesize following fix rule match javac var atch class names var atch accepted annotation processing explicitly requested str javac var first part rule symbol command error message binds input strings corresponding variables variables used second part rule produce output case var expression extracts complete string associated var start index end index denotes identity string extraction prepends string beginning appends string end contributions summary ixit declarative language encoding rules map command error message possible fixed commands sound complete polynomial time synthesis algorithm based lazy algebra synthesizing ixit rules examples following scenario another common one novice java programmers analysis formal properties ixit language synthesis algorithm java could find load main class java run qualitative quantitative evaluation synthesis algorithm benchmarks obtained xxx online motivating examples given example another similar one faq synthesizes following rule first present main ideas behind faq using concrete examples example rules presented section actual rules appearing xxx system extracting substrings match java var atch could find load main class var atch str javac var adding missing file extension java programmers particular novice ones likely encounter error accidentally pass class name instead source code file javac compiler rule extracts substring input file name starting index ending index index end string order remove extension javac employee class names employee accepted annotation processing explicitly requested often error provided javac compiler invoked file proper extension seasoned programmer would immediately recognize problem add extension end input file extracting complex substrings user trying move picture one location another got following error message rename file directory mkdir mary javac hand novice programmer search web hope finding way address error understand apply setting goal faq automatically synthesize simple fixes like one examples provided experienced users use synthesized fixes help novice users error cryptic novice users guide towards actual problem missing directory given example another similar one faq synthesizes following rule fix expr pos expr variables constants string character match var atch var atch rename var atch file directory str mkdir pos var str var var fix rule input cmd input error output cmd match expr second expression output extracts directory substring starts index ends index first occurrence character rule also adds string end extracted string pipe two output commands cmd err match cmd err var atch str var pos pos integer figure syntax rule description language ixit command repair language ixit describe ixit language expressing repair rules syntax semantics ixit presented figure figure respectively language ixit designed expressive enough capture rules found xxx online time concise enough enable efficient learning examples indices resulting respectively evaluating position expressions string given string expression substr denotes string ajl ajr undefined value otherwise notice unlike previous languages ixit allow binary recursive concatenation one key features enables polynomial time synthesis general structure ixit program rule form match cmd err positions expressions position expression one following types expressions takes input command error either produces fixed command undefined value inputs lists strings obtained extracting strings appearing input command error message respectively output fix produced rule also list strings assume inputs outputs lists strings contain space characters rule three components constant position expression pos denotes dex positive index negative expression evaluates evaluated starting index substring evaluated denotes length string example function pos pos var file first constant position evaluates index second constant position evaluates index list match expressions cmd used pattern match input command list match expressions err used pattern match input error message symbolic position expression pos denotes result applying offset index occurrence character positive result applying offset index last occurrence character negative example given string expression pos denotes index two positions first dot expression pos denotes index two positions last dot operator novel express operations supported previous work particular flashfill allows extraction exact position character positions proximity despite additional capability ixit programs synthesized polynomial time list fix expressions used produce new fixed command match expressions match expression either form denoting constant string form var atch var atch expression denotes variable index requires matched string start prefix end suffix assume two variable expressions appearing match expression index list match expressions applied list strings length generates partial function assigns variables appearing match expressions corresponding strings input example evaluating comparison flashfill dsl ixit consists match expressions original command error message perform unification variables strings form matching unification expressible flashfill use learn fix rules directly however use flashfill subroutine learn string transformations corresponding expressions similar expressions faq however flashfill dsl two major limitations support offsets regular expression matches computing position expressions contrast ixit pos operator finite token set regular expressions support constant character tokens moreover described subsection section operator yields synthesis algorithm operates polynomial time var atch var atch list strings produces function hand evaluating yields match required suffix var atch fix expressions fix expression either form str denoting constant output string form var denoting function applied string matched variable var function outputs string denotes string concatenation operator substr unify cmd unify err otherwise unify always undefined function otherwise undefined every var atch otherwise jmatch cmd err ixk jfn str kfun pos var kfun substr jpl kpos jpr indices pos kpos pos kpos indices otherwise indices figure semantics command repair dsl ixit substring expressions suk sui form var intuitively replace set one fix expressions contains obtain ixit rule contains elements symbolic representation models programs using expression size number examples enables algorithm scale large number examples recursive binary concatenate expressions flashfill dag intersection based synthesis algorithm exponential number examples synthesizing rules examples section first describe algorithm synthesizing single ixit rule set examples concrete command fixes describe partitioning algorithm learning multiple ixit programs large undifferentiated set command repair examples algorithm learning single ixit rule described figure takes input list examples example triple form outputs symbolically represented set ixit rules consistent every rule example rule outputs input algorithm processes one input example time processing first examples algorithm computed set rules consistent end algorithm outputs one rules use denote undefined result point algorithm returns means ixit rule consistent given set examples lazy rule representation core element algorithm lazy representation rules represents match fix expressions constants long new example shows parts rule constants helps reduce number variable expressions turn reduces number substring expressions considered first illustrate idea concrete example let say given two examples shown figure processing first example algorithm synthesizes ixit rule figure every match expression every fix expression constant however since seen one example yet know whether expression appearing match actually variable match expression whether element fix expression actually function variable main idea possibilities still recovered new example shows indeed variable needed using idea maintain expression constant new example shows expression actually constant exactly happens processing input example figure point order find rule consistent examples need introduce variable match second expression command match function application second element fix algorithm applies following operations previously computed rule symbolic representation multiple rules since exponentially many rules consistent input examples adopt symbolic representation set guaranteed always polynomial size synthesis algorithm takes input list examples produces output symbolic rule form match cmd err ixes cmd err tuples expressions consist either constants variables ixes list expressions symbolically represents set outputs consistent examples formally ixes either constant expression str set match expressions constants promoted variable match expressions making sure variable names unique match longest shared prefix suffix previously seen values position following table illustrates idea case try java could find load main class java run java could find load main class java meta first example second example match java var atch could find load main class var atch pos pos var pos pos var str java pos pos var match java could find load main class str java str run symbolic rule representation synthesized first example symbolic rule representation synthesized examples figure two input examples symbolic rules synthesized processing var denotes string list obtained concatenating command error input lists unify command figure matching part already computed rule figure rule java java java var atch lazy pattern matching function indvariables given rule new example iterates input components new example outputs set variables necessary match new example respect previously computed symbolic rule function ynth given rule list examples individually synthesizes components symbolic output fix expression consistent ynth incremental sense tries minimally change original fix expression fix expressions constants promoted expressions consistent current examples allowed use variables appearing match expressions second rule figure reflects update figure also shows multiple expressions represented symbolically set describe components detail next section component fix expression stant string consistent new example left unchanged case output substring operation synthesis algorithm function ynth ubstrings used compute possible expressions consistent set examples given list input examples function ynth rules uses first example function onst rule generate symbolic rule composed constant operators iteratively refines rule remaining examples shown figure second operation done function fine rule takes input symbolic rule one new example list examples every concrete rule represented behaves correctly efine rule executes two main steps using following functions given list examples function ynth ubstrings first synthesizes expressions consistent using function ubstrings runs synthesized expression examples remove inconsistent ones allsubstrings figure omits formal definition function ubstrings due space limitations describe main components given example set variable names index corresponding element output sequence trying synthesize ubstrings computes set substring expressions form var consistent result applying string string let assume length longest string appearing three lists input example compute set ubstrings iterate variable indices variable index following indvariables tries unify inputs corresponding match expressions cmd err symbolic rule generates new variable match expressions contains matching expression corresponding component example string different case var atch expression generated new variable name longest prefix suffix shared respectively indvariables presented new constant match expression promoted var atch expression prefix suffix updated accordingly indvariables determines longest prefix suffix respectively consistent appropriate component new example generates var atch extract string corresponding variable var iterations ynth uses variables computed previous step synthesize possible fix expressions consistent list examples order simplify variable naming guarantee unique names variable index corresponding element string substring compute possible pairs indices substr possible substrings possible ways place substring fixes consistent examples expressions depend variable fix component latest example fixes computed previous examples also passed input fun ynth return str str else ynth ubstrings return consistent input examples fun ynth rules onst rule refine example efine rule return rule consistent one example fun onst rule scn sem sfl cstcmd scn csterr sem cstf str str sfl return match cstcmd csterr cstf rule make consistent one example fun efine rule match cmd err ixes indvariables cmd indvariables err ynth ixes return match expressions consistent examples appear position fix expression fun ynth ubstrings ubstrings let var variable eval else variable eval return variables necessary match example fun indvariables input length match length return case case pref ongest ommon refix suf ongest ommon uffix newid create new variable var atch newid pref suf newid case var atch pref ongest ommon refix suf ongest ommon uffix var atch pref suf return expressions consistent one example appear position fix expression fun ubstrings return var evaluated output figure algorithm synthesizing ixit rules concrete examples pos size reduces synthesis algorithm still sound complete fragment ixit last restriction language capture rules interested notice analysis still holds extreme case input matches variable expressions form var atch experiments commands performance uncommon induced substring operations heterogeneous strings yield many possible implementations consider following two examples resp compute every position expression resp evaluating resp produces index resp possible positions possibilities yield expression var substr ubstrings produces set expressions worst case size restrict offset component range values symbolic expressions aaaa aaaa aaaa aaaa note efficiency implementation contingent specific choice data structure algorithms naive solution based set intersection like algorithm flashfill may experience exponential number examples due quadratic nature intersection operations bbbb bbbb bbbb bbbb performing synthesis pair examples yields pattern match consisting four var atch expressions due uniformity input strings synthesis yields possible expressions particular desired fix generated four strings supplied scmd serr four strings three substrings length yields desired output substring four pairs pos values supply appropriate indices pair two positive indices pair two negative indices two pairs consisting one positive one negative index key point point ready explain match expressions kept constants long possible processing set examples expression cmd err form every input example value component string therefore even replace expression variable instantiations values consequently every function form var produce constant output making equivalent constant function str synthesis ixit learn repair rules set examples corresponding specific incorrect use command practice however may difficult present ixit collection neatly curated sets examples ixit learns single symbolic rule process instead envision learning ixit rules undifferentiated sets examples submitted many users facilitate propose simple partitioning strategy consequence ixit structure symbolic rule matches pair command error strings fixed number tokens upon match ixit generates repaired command fixed number tokens conversely every symbolic ixit expression must learned set example triples form scmd serr lengths scmd serr vary given undifferentiated set examples sse partition disjoint subsets every property scmd serr divides subsets possible synthesize ixit rules dividing may still case individual sets contain examples representing distinct command repair rules share triple scmd serr lengths point attempt find smallest set rules synthesized examples search ranges partitions set enumerate partitions ascending order size starting ending partition consisting entirely singleton sets given partition attempt synthesize symbolic ixit rule set stopping generated rule show section expensive procedure feasible using novel contribution lazy vsa java could find load main class java run java could find load main class java test composer pkg mean one composer composer hptt mean one http html composer http rename file directory mkdir mary rename file directory mkdir bob figure examples requiring one ixit rule exists ixit program consistent examples set ixit program describe transformation consistent examples therefore proceed iteratively partitioning set attempting find partition synthesize rule every clearly initial partition yielding rule section first partition generate ixit rule every element yields rule section yields simple substitution rule described previously succinct representation version symbolic rule presented already able store exponentially many concrete ixit rules polynomial space section discuss improvements make representation succinct avoid redundancy set fix expressions enumerated function ubstrings last three components expression var often repeated many times looking figure see synthesized functions applied variable var define data structure representing sets fix expressions avoids repetitions set fix expressions represented symbolically using partial function set set position expressions formally given variable index two strings set symbolically represents set fix expressions var function efficiently implemented avoids redundancy considering example rule figure fix expressions second component output succinctly represented function defined input pos pos pos pos pos pos example partitioning consider example set shown figure first two examples set correspond repair section last two correspond repair section third fourth example correspond rule outputs composer followed token index input first token suggested error message first group examples based length components first four examples two remaining examples thus obtain two groups avoid example representation synthesized rule sense coupled set examples used synthesize present data structure keeps track important parts input examples therefore allows discard example processed modify symbolic rule representation follows given set examples cmd ynth rules run command error examples provided ynth rules synthesis obtain fix originally provided example theorem soundness given list examples every rule con synthrules every example jrk every variable var atch match component becomes pair var atch second component list strings binds var input components proof sketch repeated applications indvariables promote expression new example match moreover refining var atch indvariables chooses longest prefix suffix consistent examples seen far thus cmd err correctly match examples soundness resulting derives fact iteration loop ynth rules invocation ynth ubstring ynth takes account examples seen previous iterations loop moreover invocation begins set possible expressions prunes inconsistent example seen far since parts match expressions promoted variables input examples show required synthesis algorithm explicitly keep track possible rules consistent examples completeness result reflects idea every set fix expressions represented function corresponding component fix expression becomes pair second component list strings appear position output components using data structure need store examples always symbolic rule representation concrete outputs taking account updated data structures algorithm ynth rules returns symbolic rule form match cmd err ixes cmd err lists expressions form var ixes list expressions form str data structure representing multiple fix expressions set concrete ixit rules induced symbolic representation following theorem completeness given set examples every fixit rule consistent either con synthrules exists example con synthrules con match cmd err match cmd err con con con con str str con var proof sketch concretely particular rule consistent might appear ynth rules however happen match expression variables match rule fixed providing example forces algorithm promote variables required match expressions formal properties properties language fixit define size input example size rule consistent sum lengths size size set examples exponentially many rules consistent study formal properties synthesis algorithm language ixit specific properties describe behavior synthesis algorithm absence partitioning strategy described section properties synthesis algorithm first synthesis algorithm invariant respect order training examples presented thus properties symbolic rule generated ynth rules discussed solely terms set examples provided ynth rules theorem number consistent rules given set examples size set con synthrules contains poly rules proof sketch showed section position output ixit rule potentially polynomially many expressions consistent provided examples ith position free choose possible lrs independently choice positions thus number possible rules potentially exponential despite exponential number rules represented data structure allows ynth rules encode rules using polynomial size next exists active learning algorithm learning ixit rules requires polynomial number queries user theorem order invariance given list examples every permutation examples con synthrules con synthrules proof sketch consider list examples two examples differ ith position respective commands ith expression cmd promoted var atch regardless order presented ynth rules moreover strings ith position commands share prefix suffix reordering change fact thus discovered var atch expressions vary based order holds err promotion constants ynth moreover ynth ubstring starts scratch iteration loop ynth rules depends output ynth final iteration since variable set vary based ordering final invocation ynth depend ordering second synthesis algorithm produces rules consistent input examples select arbitrary concrete rule set specified symbolic rule generated theorem complexity active learning exists target rule size exists active learning algorithm learn asking poly queries form output input proof sketch algorithm first asks queries figure match expressions variables ones constants output component exists two possible fix expressions consistent examples asks query differentiates two since kni many expressions output algorithm ask polynomially many queries present qualitative quantitative analysis algorithm assess expressiveness faq attempting synthesis benchmark suite includes rules tool xxx evaluate performance faq scalability implementation evaluation describe implementation details faq well experimental evaluation faq set examples test cases isolated xxx web forums implementation implemented language ixit synthesis algorithm system called faq faq implemented consists additional optimizations design choices described implementation optimizations function ubstrings figure synthesizes functions consistent first pair strings example set applies synthesized functions elements filtering consistent functions practice first compute longest common prefixes suffixes strings appearing components avoid enumerating instances form prefixes suffixes output string appearing optimization based following property efine rule function adding new example function indvariables introduces new set variables new instances already appear depend one newly introduced variables based idea function ubstrings compute functions form var reuse previously computed functions variables simply filtering ones behave correctly newly introduced example number examples observed natural provide two five examples per benchmark faq uniquely learn desired fix rule also provided additional examples manually testing learned rules yielding set three six examples given rule appearing figure example used two examples figure another example file name future envision users contribute different examples system automatically building large corpus learned fix rules examples synthetic examples reverse engineered sources also natural examples exercise range path file names one would expect see real unix system case repaired command section natural set would consist two distinct directory names share prefixes suffixes well filenames distinct prefixes extensions ranking since multiple possible expressions ixit consistent examples employ simple ranking technique select expression amongst multiple expressions generate desired output string select expression uses variable lowest leftmost one similarly included var atch expressions implicitly encode rules matching prefixes suffixes respectively choose longest others example set increases size envision users likely submit diverse sets examples particularly use cases thousands users submitting examples users submit examples draw heterogeneous collections command parameters var atch prefixes suffixes converge least restrictive versions similarly faq discover least restrictive set constants match expressions input parameters vary set examples spurious ambiguities eliminated faq presented specific fix examples function counterexamples unnecessary substring expressions benchmark suite compiled benchmark suite initial set benchmarks collected xxx online help forums considered repair rules hardcoded xxx tool assess expressiveness faq since rules xxx use arbitrary python code hard exactly compare ones produced faq use manual testing check rule generated tool consistent rule xxx manually constructed set examples based textual substitutions performed xxx rules sixteen example sets obtained examples found survey help forums web commands consist various types git svn mvn commands including committing reverting deleting repositories well installing removing packages faq system able synthesize rule benchmarks remaining failing benchmarks divided three broad categories operations searching specific strings context complex patterns checking relationships variable expressions iii error messages displaying parts input file content provide examples rules elaborate categories section qualitative evaluation given single set containing examples cases faq capable synthesizing rule performed synthesis described section rule retained single example training set used test accuracy rule also report often given input could repaired using one rule results cases faq synthesized rule consistent corresponding xxx rule web forum answer cases synthesize one ixit rule capture different possible behaviors single rule xxx example one try adding sudo front command several possible errors command found permission etc cases thanks partitioning algorithm faq generated separate rule possible error message case synthesized rule correctness evaluation describe experimental evaluation experiments run intel core cpu ram implementation made publicly available review process eager lazy vsa size expressions cmdsize errsize synthesis time econds distribution cmd err fix expr size fixsize eager vsa lazy vsa enumerating partitions size largest group benchmarks figure synthesis times different benchmarks lazy rule representations figure distribution benchmarks terms individual sizes expressions examples evaluation lazy vsa synthesis time figure show time taken partition synthesize ixit rules benchmarks using lazy rule representation number examples per benchmark increases nonlazy representation always considers match fix expressions variables rather initially starting constants test performance lazy representations size input set increases iteratively increase size training set test add single example one benchmarks attempt synthesis plot synthesis time respect largest set examples must enumerate possible partitions successfully synthesize rules understand performance overhead induced synthesis also evaluate version algorithm enumerates partitions without performing synthesis algorithm iteratively increased training set size algorithm reached second timeout vsa incurs significant overhead scales much worse lazy version reaching timeout largest set examples lazy vsa contrast much closer optimum synthesis time negligible compared inherent cost enumerating partitions set fact lazy synthesis actually completes faster exhaustive enumeration reasonable first partitioning yields successful ixit rule subsets tends somewhere near middle enumeration thus incur cost enumerating remainder search space summary lazy vsa strictly outperforms vsa handle much larger sets examples independent choice examples correct rule synthesized synthesized regardless subset examples provided rule selected distribution rule sizes define size expression number strings present distribution size benchmarks terms sizes tuples examples shown figure note show two benchmarks graph disproportionately high expression size clarity average total size examples benchmarks maximum size average sizes individual expressions examples maximum maximum iii maximum distribution rule matching set example provided individual rule isolated one example measure accuracy tool test examples correctly described least one synthesized rules majority test cases exactly one rule matched command error message remaining test cases matched multiple rules came collections example sets represented different fixes command error messages total test cases test cases matched one rule test cases matched multiple rules expression synthesized rules distribution str expressions synthesized ixit rules shown figure output components synthesized rule contain average expressions concretely synthesized rule contains average expressions ranking consistent hypothesis section diverse set examples sufficient eliminating spurious restrictions substring expressions every test case rule chosen ranking policy capable correcting test cases presented practice many rules still several possible correct expressions however remaining ambiguity occurs string appear many times command error message string employee example section synthesis time number expressions synthesis time different numbers expressions repair rule shown figure expected benchmarks contain expressions take negligible time benchmarks involving two expressions average require time benchmarks single expression interestingly benchmarks expressions take lesser time benchmarks expressions possible explanation behavior complexity substring extraction tasks benchmarks relatively simpler identity benchmarks expressions quantitative evaluation report quantitative metrics synthesis algorithm section report data benchmarks faq successfully synthesize ixit rule use denote average standard deviation fstr synthrule time example exprs sublr learning time size fix expression exprs fix exprs benchmarks number expressions examples figure distribution str expressions final synthesized repair expression figure synthesis times increasing size examples input matching ixit also check whether string input command repeated learning time synthrules time sublr exprs eight rules operations searching context file system configuration file etc specific strings complete output ixit receives inputs command error message rules currently use context algebra synthesis concept versionspace algebra vsa first introduced mitchell context machine learning later used lau learn programs demonstrations programming system called smartedit since used many pbe systems various domains including syntactic string transformations flashfill table transformations number transformations text extraction text files flashextract transformation spreadsheets relational tables flashrelate synthesis algorithm also uses vsa succinctly represent large set conforming expressions however contrast previous approaches represent conforming expressions concretely use intersection refinement synthesis algorithm maintains lazy representation rules concretizes choices demand lazy fashion needed moreover careful design dsl operators corresponding vsa faq lead polynomial time synthesis algorithm unlike previous approaches exponential time synthesis algorithms particular illustrative compare flashfill dsl ixit like ixit flashfill synthesizes string manipulations examples specific performance properties make less suitable large scale learning large sample sets prior developing faq evaluated possibility simply using flashfill algorithm purpose learning command repair rules early empirical results indicated algorithm scaled poorly error messages increased length common occurrence benchmarks moreover limitations offset operator position expressions support finite regular expression tokens made flashfill unsuitable learning expressions isolated several theoretical properties flashfill key operators yielded poor performance large inputs particular binary concatenation operator arbitrary substrings number sublr expressions fix expr figure synthesis times varying number expressions repair rule scalability synthesis algorithm example size since examples collected relatively small size maximum size strings evaluate scalability ynth rules algorithm creating artificial examples increasing sizes create artificial examples repeating commands multiple times example shown section synthesis times increasing size examples shown figure graph observe synthesis times scale quadratic fashion respect example size related work limitations showed language ixit able express many realworld command line repair rules rules synthesized using examples present limitations approach particular respect xxx rules ixit could describe complex patterns two rules checking complex properties input ixit capture example ixit check whether error message contains special character ixit conditional matching limited whole string matching thus check file name contains whitespace character character relative logic occurs substring generation xxx provides python interface command substitution repair rules requires degree language toolspecific knowledge may accessible command line novices particularly substring operations required derive desired command much like flashfill aim emulate workflow users communicating experts web forums beginner learning command line python string manipulations likely fairly challenging task cost incorrectly transformed shell command potentially catastrophic situation desires new xxx rule user may provide example several pairs desired fix expert would write desired python code nofaq shortens loop moving fix synthesis polynomial time algorithm user machine entire input string induces dag structure symbolic representation programs explicitly given example output string exists node position edge represents substring edge labelled set functions example inputs yield substring thus path represents concatenation output several string operations yields desired output given dag consistent set examples flashfill incorporates new example represented dag taking cartesian product vertices construct new dag edge label set new dag represents set functions part correct program examples also map inputs substring output iterated cartesian products yield time complexity exponential number examples ixit contrast posesses unary string operations constrained specific variable terms ixit unary operator yields language disjoint flashfill concatenation removed obtain language expressive enough large set practical command repair transformations isolated real use cases dramatically improving performance constrained nature program representation lets synthesis algorithm eliminate programs inconsistent new example without directly computing intersection two sets candidate programs ensuring performance even worst case moreover ixit also allows repair transformations require arbitrary offsets regex match expressible flashfill dsl rule learning rules provide simple way represent programmers actions general type data transformation rule learning extensively investigated classical machine learning data mining goal rule learning discover mine rules describing interesting relations appearing data common concept classes describing rules horn clauses association rules approach presented paper differs rule learning two aspects rules expressed complex concept class hard learn ixit programs examples given teacher mind target rule future plan build system uses rule learning techniques mine ixit rules unsupervised data programming examples pbe pbe active research area hci communities long time addition data wrangling pbe techniques recently developed various domains including interactive synthesis parsers synthesis recursive functional programs algebraic data types synthesizing sequence program refactorings imperative data structure manipulations network policies technique also learns repair rules examples buggy fixed commands problem domain learning command repairs learning techniques using lazy vsa quite different pbe systems program synthesis resurgence program synthesis research recent years addition examples described several techniques developed handling forms specifications partial programs reference implementations concrete traces specification mechanisms found useful several domains believe examples natural mechanism specifying command line repairs especially beginners also recent movement towards using techniques synthesis pliny project http future envision system also make use large number examples buggy commands corresponding repairs learn big database ixit rules program repair research automated program repair focuses automatically changing incorrect programs make meet desired specification main challenge efficiently search space programs find one behaves correctly prominent search techniques enumerative genprog uses genetic programming repeatedly alter incorrect program hope make correct approaches use large amount code publicly available online synthesize likely changes input program prophet patch generation system learns probabilistic model correct code set successful human patches unlike techniques learn global model code repair across different applications technique learns repairs observing expert users fix buggy commands incorrect command user started together error message correct command wrote fix conclusion future directions presented tool faq suggests possible fixes common buggy commands learning examples experts fix issues language design walks fine line expressivity performance careful choice unary operators variables exclusion arbitrary substring operations avoid worst case performance still maintaining useful degree functionality faq able instantly synthesize rules appearing popular repair tool xxx rules online help forums although faq tool aimed towards repairing commands believe novel combination synthesis program repair quite general applicable many domains well plan apply methodology complex tasks correcting syntax errors source code applying code optimization editing configuration files future hope create tool take large command histories expert users quickly derive rules well synthesize new rules online experts use shell crowdsourced repairs helpmeout social recommender system helps novice users facing programming errors showing examples programmers corrected similar errors helpmeout show examples similar fixes concretely show user code corrected aspect major difference helpmeout faq references ence software engineering icse pages piscataway usa ieee press isbn url http alur dallal fisman garg juniwal kressgazit madhusudan martin raghothaman saha seshia singh torlak udupa synthesis dependable software systems engineering pages leung sarracino lerner interactive parser synthesis example proceedings acm sigplan conference programming language design implementation portland usa june pages doi url http barowy gulwani hart zorn flashrelate extracting relational data spreadsheets using examples proceedings acm sigplan conference programming language design implementation pldi pages new york usa acm isbn doi url http lieberman wish command programming example morgan kaufmann long rinard automatic patch generation learning correct code proceedings annual acm sigplansigact symposium principles programming languages popl pages doane mcnamara kintsch polson clawson prompt comprehension unix command production memory cognition mitchell generalization search artif feser chaudhuri dillig synthesizing data structure transformations examples proceedings acm sigplan conference programming language design implementation portland usa june pages doi url http osera zdancewic program synthesis proceedings acm sigplan conference programming language design implementation portland usa june pages doi url http furnkranz gamberger lavrac rule learning nutshell foundations rule learning cognitive technologies pages springer berlin heidelberg isbn doi url http discovery analysis presentation strong rules frawley editors knowledge discovery databases pages aaai press raychev sridharan vechev refactoring synthesis proceedings acm sigplan international conference object oriented programming systems languages applications oopsla part splash indianapolis usa october pages doi url http goues forrest weimer current challenges automatic software repair software quality journal issn doi url http gulwani automating string processing spreadsheets using inputoutput examples proceedings annual acm sigplansigact symposium principles programming languages popl pages new york usa acm isbn doi url http raychev vechev yahav code completion statistical language models proceedings acm sigplan conference programming language design implementation pldi pages new york usa acm isbn doi url http gulwani harris singh spreadsheet data manipulation using examples commun acm issn doi url http raychev vechev krause predicting program properties big code proceedings annual acm symposium principles programming languages popl mumbai india january pages doi url http harris gulwani spreadsheet table transformations examples proceedings acm sigplan conference programming language design implementation pldi san jose usa june pages doi url http schkufza sharma aiken stochastic superoptimization architectural support programming languages operating systems asplos houston usa march pages doi url http hartmann macdougall brandt klemmer would programmers suggesting solutions error messages proceedings sigchi conference human factors computing systems chi pages new york usa acm isbn doi url http singh gulwani synthesizing number transformations examples proceedings international conference computer aided verification cav pages berlin heidelberg isbn doi url http lau wolfman domingos weld programming demonstration using version space algebra mach issn doi url http singh gulwani learning semantic string transformations examples proc vldb apr issn doi url http gulwani flashextract framework data extraction examples proceedings acm sigplan conference programming language design implementation pldi pages new york usa acm isbn doi url http singh synthesizing data structure manipulations storyboards sigsoft fse pages program synthesis sketching phd thesis eecs berkeley rabbah bodik ebcioglu programming sketching programs pldi goues forrest weimer systematic study automated program repair fixing bugs proceedings international udupa raghavan deshmukh martin alur transit specifying protocols concolic every compf holds say ixes partially complete respect compf ixes snippets acm sigplan conference programming language design implementation pldi seattle usa june pages doi url http yakout elmagarmid neville ouzzani ilyas guided data repair proc vldb issn doi url http yuan alur loo netegg programming network policies examples proceedings acm workshop hot topics networks los angeles usa october pages doi url http definition exists comp cmd err say complete respect position compf iff compf ind tind fstr iff following properties hold every ind jtind kfun exists function every ind jtkfun exists proofs theorems first define notion completeness symbolic rule intuitively symbolic rule summarize possible correct rules complete definition fix completeness every compf holds say ixes complete respect compf ixes definition completeness let match cmd err ixes definition rule completeness comp cmd err compf ixes say complete compr symbolic rule cmd err ixes sequence examples say cmd complete produces variables compc cmd iff every notice permutation compr iff compr proposition compr con let sequence examples match cmd err ixes symbolic rule compr every concrete rule con consistent example moreover every rule str iff every example iff exists two examples every example prefix moreover longest prefix every example suffix moreover longest suffix match cmd err consistent cmd following true every fstr iff con proof immediate definition definition con specifically assume con consistent example way inconsistent match expression cmd err fixed string equal string definition precludes possibility similarly follows definition one fix expressions constant string consistent every example fix definition set expressions one consistent every example definition completeness analogously say err complete produces variables compe err iff every str iff every example iff exists two examples every example prefix moreover longest prefix every example suffix moreover longest suffix show ynth ubstring ynth intended behaviour respect definition lemma correctness ynth ubstring let cmd sequence examples every length set variables index output fix exist synthsubstring iff comp definition input completeness compc cmd compe err hold say cmd err complete produce variables comp cmd err proof lemma states ynth ubstring returns functions consistent given examples proof induction length base case follows definition ubstrings inductive step also trivial ynth ubstring simply runs functions computed far added example filters consistent since varaible correct definition partial say partially complete compf respect following condition holds fstr iff every example lemma correctness efine rule let symbolic rule sequence examples example refinerule compr compr beginning loop remains correct notice order examples matter lemma correctness ynth let list strings sequence symbolic fix expressions proof immediate definition lemmas cmd lemma correctness ynth rules let sequence examples synthrules compr sequence examples every length example set variables compf synthfix ixes iff compf ixes proof induction length case result onst rule clearly satisfies compr inductive step follows lemma conclude proofs theorems theorem follow order invariance compr proposition lemmas proof immediate induction using lemma step ynth iterates string example fix compares corresponding fix expression see interesting case proof either fixed string equal correpsonding set expressions case correctness follows directly lemma ynth calls ynth ubstring refine set substring operation generating well values proof theorem let cmd sequence examples ynth rules match cmd err ixes symbolic rule match concrete rule consistent examples belongs con done assume case cmd err proposition exists tik tiv form pvl pvr var output function input examples captured using constant output case enough create new example starting example modify sjv pvl pvr var returns value different previous one done simply adding new character pvl pvr replaced result applying modified input case cmd err means uses variables notice definition set variables used necessary fixed changing input example variable var cmd err replace string symbol appearing replaced result applying modified input rule still con modify example using techniques case since cmd err next show correctness efine rule indvari ables lemma correctness indvariables let cmd sequence examples cmd err two sequences match expressions lengths respectively exists two sets variables compc cmd compe err findvariables cmd cmd compc err findvariables err err compe proof two statements proved separately proof identical proofs induction length first argument indvariables simple case analysis following definitions consider case cmd run indvariables cmd changed nth iteration loop cmd equal cmd case indvariables introduces new variable match expression var atch respectively longest shared prefix suffix cmd cmd command complete follows directly definition command completeness continues hold var atch cmd changed iteration longest common prefix suffix shared cmd equal case var atch replaced var atch var atch equal common prefix suffix longest prefix suffix shared first values clearly longest prefix suffix shared first values moreover cmd satisfied criteria command completeness thus follows command completeness continues hold clear determining shorteset prefix suffix shared every depend order examples presented
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learning compose neural networks question answering jacob andreas marcus rohrbach trevor darrell dan klein department electrical engineering computer sciences university california berkeley jda rohrbach trevor klein jun abstract describe question answering model applies images structured knowledge bases model uses natural language strings automatically assemble neural networks collection composable modules parameters modules learned jointly parameters via reinforcement learning world question answer triples supervision approach term dynamic neural module network achieves results benchmark datasets visual structured domains cities georgia atlanta module inventory section lookup relate find find relate lookup georgia network layout section knowledge source find city city montgomery relate georgia lookup georgia atlanta figure learned syntactic analysis used assemble collection neural modules deep neural network applied world representation produce answer introduction paper presents compositional attentional model answering questions variety world representations including images structured knowledge bases model translates questions dynamically assembled neural networks applies networks world representations images knowledge bases produce answers take advantage two largely independent lines work one hand extensive literature answering questions mapping strings logical representations meaning series recent successes deep neural models image recognition captioning constructing neural networks instead logical forms model leverages best aspects linguistic compositionality continuous representations model two components trained jointly first collection neural modules freely composed figure second network layout predictor assembles modules complete deep networks tailored question figure previous work used modular structures visual learning andreas learn network structure predictor jointly module parameters extend visual primitives previous work reason structured world representations training data consists world question answer triples approach requires supervision network layouts achieve performance two markedly different question answering tasks one questions natural images another compositional questions united states deep networks functional programs begin discussion kinds composed networks would like learn released code http andreas describe heuristic approach decomposing visual question answering tasks sequence modular example question color bird might answered two steps first bird figure second color part image figure first step generic module called find expressed fragment neural network maps image features lexical item bird distribution pixels operation commonly referred attention mechanism standard tool manipulating images text representations hermann first contribution paper extension generalization mechanism enable reasoning structured semantic representations figure shows module used focus entity georgia grounding domain generally representing every entity universe discourse feature vector obtain distribution entities corresponds roughly logical denotation obtained distribution existing neural approaches use immediately compute weighted average image features project back labeling describe module figure logical perspective suggests number novel modules might operate attentions combining analogy conjunction disjunction inspecting directly without return feature space analogy quantification figure modules discussed detail section unlike formal counterparts differentiable facilitating integration learned models building previous work learn behavior collection heterogeneous modules world question answer triples second contribution paper model learning assemble modules compositionally isolated modules limited obtain expressive power comparable either formal approaches monolithic deep networks must composed larger structures figure shows simple examples composed structures realistic tasks even larger black white true describe exists color montgomery georgia atlanta find bird find state montgomery georgia atlanta figure simple neural module networks corresponding questions color bird states neural find module computing attention pixels operation applied knowledge base using attention produced lower module identify color region image attended performing quantification evaluating attention directly works required thus goal automatically induce computation descriptors use familiar functional notation formal semantics liang represent write two examples figure describe color find bird exists find state respectively network layouts specify structure arranging modules lexical parameters complete network andreas use rules deterministically transform dependency trees layouts restricted producing simple structures like data full generality need solve harder problems like transforming cities georgia figure find city relate lookup georgia paper present model learning select structures set automatically generated candidates call model dynamic neural module network note unlike formal semantics behavior primitive functions unknown related work extensive literature database question answering strings mapped logical forms evaluated execution model produce answers supervision may provided either annotated logical forms wong mooney kwiatkowski andreas world question answer triples alone liang pasupat liang general set primitive functions logical forms assembled fixed one recent line work focuses inducing new predicates functions automatically either perceptual features krishnamurthy kollar underlying schema kwiatkowski model describe paper unified framework handling perceptual schema cases differs existing work primarily learning differentiable execution model continuous evaluation results neural models question answering also subject current interest include approaches model task directly multiclass classification problem iyyer models attempt embed questions answers shared vector space bordes attentional models select words documents sources hermann approaches generally require answers retrieved directly based surface linguistic features without requiring intermediate computation structured approach described yin learns query execution model database tables without natural language component previous efforts toward unifying formal logic representation learning include grefenstette krishnamurthy mitchell lewis steedman beltagy component work relies recent advances convolutional networks computer vision simonyan zisserman particular fact late convolutional layers networks trained image recognition contain rich features useful vision tasks preserving spatial information features used image captioning visual yang previous approaches visual question answering either apply recurrent model deep representations image question ren malinowski use question compute attention input image answer based question image features attended yang saenko approaches include simple classification model described zhou dynamic parameter prediction network described noh models assume fixed computation performed image question compute answer rather adapting structure computation question noted andreas previously considered simple generalization attentional approaches small variations network structure permitted structure chosen deterministic syntactic processing questions approaches general family include universal parser sketched bottou graph transformer networks bottou neural networks towell shavlik recursive neural networks socher use fixed tree structure perform linguistic analysis without external world representation unaware previous work simultaneously learns parameters structures networks model recall goal map questions world representations answers process involves following variables world representation question answer network layout collection model parameters model built around two distributions layout model chooses layout sentence execution model applies network specified ease presentation introduce models reverse order first imagine always observed section describe evaluate learn modules parameterized within fixed structures section move real scenario unknown describe predict layouts questions learn jointly without layout supervision evaluating modules modules used paper shown names type constraints first row description module computation following lookup given layout assemble corresponding modules full neural network figure apply knowledge representation intermediate results flow modules answer produced root denote output network layout input world jzkw explicitly referencing substructure alternatively write module submodule outputs define execution model jzkw world representation represented collection vectors expressed matrix nonlinearity denotes rectified linear unit assumes root module produces distribution labels set possible layouts restricted module type constraints modules like find operate directly input representation others like describe also depend input specific earlier modules two base types considered paper attention distribution pixels entities labels distribution answers parameters tied across multiple instances module different instantiated networks may share parameters others modules parameter arguments shown square brackets ordinary inputs shown parentheses parameter arguments like running bird example section provided layout used specialize module behavior particular lexical items ordinary inputs result computation lower network addition weights modules global weights shared across instances module shared modules write global weights weights associated parameter argument denote possibly broadcasted elementwise addition multiplication respectively complete set global weights weights constitutes every module access attention lookup produces attention focused entirely index relationship words positions input map known ahead time string matches database fields jlookup basis vector ith position elsewhere find attention computes distribution indices concatenating parameter argument position input feature map passing concatenated vector mlp find jfind softmax attention attention directs focus one region input another behaves much like find module also conditions behavior current region attention let element relate relate jrelate softmax attention attention performs operation analogous set intersection attentions analogy probabilistic logic suggests multiplying probabilities jand attention labels input attention average used predict answer representation describe describe computes weighted average jdescribe softmax attention labels existential quantifier inspects incoming attention directly produce label rather intermediate feature vector like describe jexists softmax max exists exists cities georgia city georgia relate find city lookup georgia relate find city lookup georgia relate lookup georgia figure generation layout candidates input sentence represented dependency parse fragments dependency parse associated appropriate modules fragments assembled full layouts observed model described far corresponds largely andreas though module inventory particular new exists relate modules depend spatial structure input enables generalization nonvisual world representations learning simplified setting straightforward assuming module layout describe exists module instantiated network corresponds distribution labels conditioned layouts train maximize log directly understood scheme decisions parameters tie governed observed layouts assembling networks next describe layout model first use fixed syntactic parse generate small set candidate layouts analogously way semantic grammar generates candidate semantic parses previous work berant liang semantic parse differs syntactic parse two primary ways first lexical items must mapped onto possibly smaller set semantic primitives second semantic primitives must combined structure closely exactly parallels structure provided syntax example state province might need identified field database schema states capital might need identified correct situ quantifier scope avoid structure selection problem continuous representations simplify lexical selection problem modules accept vector parameter associate parameters words rather semantic tokens thus turn combinatorial optimization problem associated lexicon induction continuous one order learn province state denotation sufficient learn associated parameters close embedding task amenable gradient descent note easy optimizability sense must still learn associate independent lexical item correct vector remaining combinatorial problem arrange provided lexical items right computational structure respect layout prediction like syntactic parsing ordinary semantic parsing rely syntactic parser get way work syntactic structure provided stanford dependency parser marneffe manning construction layout candidates depicted figure proceeds follows represent input sentence dependency tree collect nouns verbs prepositional phrases attached directly copula associate layout fragment ordinary nouns verbs mapped single find module proper nouns single lookup module prepositional phrases mapped fragment relate module preposition find module enclosed head noun form subsets set layout fragments subset construct layout candidate joining fragments module inserting either measure describe module top subset thus results two parse candidates layouts resulting process feature relatively flat tree structure one conjunction one quantifier strong simplifying assumption appears sufficient cover examples appear tasks approach includes categories relations simple quantification range phenomena considered generally broader previous work krishnamurthy kollar matuszek generated set candidate parses need score ranking problem rest approach solve using standard neural machinery particular produce lstm representation question representation query pass representations multilayer perceptron mlp query feature vector includes indicators number modules type present well associated parameter arguments one easily imagine sophisticated parsescoring model simple approach works well tasks formally question let lstm encoding question last hidden layer lstm applied input question let proposed layouts let feature vector representing ith layout score layout bhq output mlp inputs parameters finally normalize scores obtain distribution defined layout selection module network execution model ready define model predicting answers given world question pairs key constraint want minimize evaluations involves expensive application deep network large input representation tractably evaluate involves application shallow network relatively small set candidates opposite situation usually encountered semantic parsing calls query execution model fast set candidate parses large score exhaustively fact problem closely resembles scenario faced agents reinforcement learning setting cheap score actions potentially expensive execute obtain rewards adopt common approach literature express model stochastic policy policy first sample layout distribution apply knowledge source obtain distribution answers chosen train execution model directly maximizing log respect ordinary backpropagation hard selection nondifferentiable optimize using policy gradient method gradient reward surface respect parameters policy log reinforce rule williams expectation taken respect rollouts policy reward goal select network makes accurate predictions take reward identically negative execution phase log log thus update model timestep simply gradient chosen layout scaled accuracy layout predictions training time approximate expectation single rollout step update direction log log single optimized using adadelta zeiler gradient clipping norm zhou noh yang nmn sheep ear color wearing man dragging describe find sheep find ear describe color find wear describe find man tag white boat board figure sample outputs visual question answering task second row shows final attention provided input describe module first two examples model produces reasonable parses attends correct region images ear woman clothing generates correct answer third image verb discarded wrong answer produced experiments framework described paper general interested well performs datasets varying domain size linguistic complexity end evaluate model tasks opposite extremes criteria large visual question answering dataset small collection structured geography questions questions images first task visual question answering challenge vqa antol vqa dataset consists images paired questions answers figure use vqa release employing development set model selection hyperparameter tuning reporting final results evaluation server set experiments described section input feature representations computed fifth convolutional layer vggnet pooling simonyan zisserman input images scaled computing representations found performance task number table results vqa test server nmn model andreas model described paper best candidate layouts relatively simple describe find modules used layouts contain two conjuncts one weakness basic framework difficulty modeling prior knowledge answers form bears brown kinds linguistic prior essential vqa task easily incorporated simply introduce extra hidden layer recombining final module network output input sentence representation see equation replacing equation log ahq bjzkw modules output type labels understood producing answer embedding rather distribution answers allows question influence answer directly results shown table use dynamic networks provides small gain noticeably questions achieve results task outperforming highly effective visual model zhou model dynamic network parameter prediction fixed network structure noh conventional attentional model yang previous approach using neural module networks structure prediction andreas examples shown figure general model learns focus correct region image tends consider broad window around region facilitates answering questions like cat requires knowledge surroundings well object question accuracy model nmn geoqa key largo island exists lookup find island yes correct national parks florida find park relate lookup florida everglades correct table results geoqa dataset geoqa dataset quantification approach outperforms purely logical model model learned perceptual predicates original dataset fixedstructure nmn evaluation conditions beaches florida exists lookup beach relate lookup florida yes wrong parse beach city florida questions geography next set experiments consider focuses geoqa geographical task first introduced krishnamurthy kollar task originally paired visual question answering task much simpler one discussed appealing number reasons contrast vqa dataset geoqa quite small containing examples two baselines available one using classical semantic parser backed database another induces logical predicates using linear classifiers spatial distributional features allows evaluate quality model relative perceptually grounded logical semantics well strictly logical approaches geoqa domain consists set entities states cities parks participate various relations take world representation consist two pieces set category features used find module different set relational features used relate module experiments use subset features originally used krishnamurthy original dataset includes quantifiers treats questions cities texas cities texas identically interested testing parser ability predict variety different structures introduce new version dataset distinguishes two cases expects boolean answer questions second kind results shown table original work report results set lookup beach lookup city relate lookup florida none wrong module behavior figure example layouts answers selected model geoqa dataset incorrect predictions correct answer shown parentheses vironments dynamic model outperforms logical perceptual models described krishnamurthy kollar well neural module net nmn improvement particularly notable dataset quantifiers dynamic structure prediction produces relative improvement fixed baseline variety predicted layouts shown figure conclusion introduced new model dynamic neural module network answering queries structured unstructured sources information given question world answer triples training data model learns assemble neural networks fly inventory neural models simultaneously learns weights modules composed novel structures approach achieves results two tasks believe success work derives two factors continuous representations improve expressiveness learnability semantic parsers replacing discrete predicates differentiable neural network fragments bypass challenging combinatorial optimization problem associated induction semantic lexicon structured world representations neural predicate representations allow model invent reusable attributes relations expressed schema perhaps importantly extend compositional questionanswering machinery complex continuous world representations like images semantic structure prediction improves generalization deep networks replacing fixed network topology dynamic one tailor computation performed problem instance using deeper networks complex questions representing combinatorially many queries comparatively parameters practice results considerable gains speed sample efficiency even little training data observations limited question answering domain expect applied similarly tasks like instruction following game playing language generation acknowledgments supported national science foundation graduate fellowship supported fellowship within fit german academic exchange service daad work additionally supported darpa afrl dod muri award nsf awards berkeley vision learning center references jacob andreas andreas vlachos stephen clark semantic parsing machine translation proceedings annual meeting association computational linguistics sofia bulgaria jacob andreas marcus rohrbach trevor darrell dan klein neural module networks proceedings conference computer vision pattern recognition stanislaw antol aishwarya agrawal jiasen margaret mitchell dhruv batra lawrence zitnick devi parikh vqa visual question answering proceedings international conference computer vision islam beltagy cuong chau gemma boleda dan garrette katrin erk raymond mooney montague meets markov deep semantics probabilistic logical form proceedings joint conference distributional logical semantics pages jonathan berant percy liang semantic parsing via paraphrasing proceedings annual meeting association computational linguistics volume page antoine bordes sumit chopra jason weston question answering subgraph embeddings proceedings conference empirical methods natural language processing bottou yoshua bengio yann cun global training document processing systems using graph transformer networks proceedings conference computer vision pattern recognition pages ieee bottou machine learning machine reasoning machine learning marneffe christopher manning stanford typed dependencies representation proceedings international conference computational linguistics pages edward grefenstette towards formal distributional semantics simulating logical calculi tensors joint conference lexical computational semantics karl moritz hermann tomas kocisky edward grefenstette lasse espeholt kay mustafa suleyman phil blunsom teaching machines read comprehend advances neural information processing systems pages mohit iyyer jordan leonardo claudino richard socher hal iii neural network factoid question answering paragraphs proceedings conference empirical methods natural language processing jayant krishnamurthy thomas kollar jointly learning parse perceive connecting natural language physical world transactions association computational linguistics jayant krishnamurthy tom mitchell vector space semantic parsing framework compositional vector space models proceedings acl workshop continuous vector space models compositionality tom kwiatkowski luke zettlemoyer sharon goldwater mark steedman inducing probabilistic ccg grammars logical form higherorder unification proceedings conference empirical methods natural language processing pages cambridge massachusetts tom kwiatkowski eunsol choi yoav artzi luke zettlemoyer scaling semantic parsers ontology matching proceedings ference empirical methods natural language processing mike lewis mark steedman combining distributional logical semantics transactions association computational linguistics percy liang michael jordan dan klein learning compositional semantics proceedings human language technology conference association computational linguistics pages portland oregon mateusz malinowski marcus rohrbach mario fritz ask neurons approach answering questions images proceedings international conference computer vision cynthia matuszek nicholas fitzgerald luke zettlemoyer liefeng dieter fox joint model language perception grounded attribute learning international conference machine learning hyeonwoo noh paul hongsuck seo bohyung han image question answering using convolutional neural network dynamic parameter prediction arxiv preprint panupong pasupat percy liang compositional semantic parsing tables proceedings annual meeting association computational linguistics mengye ren ryan kiros richard zemel exploring models data image question answering advances neural information processing systems simonyan zisserman deep convolutional networks image recognition arxiv preprint richard socher john bauer christopher manning andrew parsing compositional vector grammars proceedings annual meeting association computational linguistics geoffrey towell jude shavlik artificial neural networks artificial intelligence ronald williams simple statistical gradientfollowing algorithms connectionist reinforcement learning machine learning yuk wah wong raymond mooney learning synchronous grammars semantic parsing lambda calculus proceedings annual meeting association computational linguistics volume page huijuan kate saenko ask attend answer exploring spatial attention visual question answering arxiv preprint kelvin jimmy ryan kiros kyunghyun cho aaron courville ruslan salakhutdinov richard zemel yoshua bengio show attend tell neural image caption generation visual attention international conference machine learning zichao yang xiaodong jianfeng gao deng alex smola stacked attention networks image question answering arxiv preprint pengcheng yin zhengdong hang ben kao neural enquirer learning query tables arxiv preprint matthew zeiler adadelta adaptive learning rate method arxiv preprint bolei zhou yuandong tian sainbayar sukhbaatar arthur szlam rob fergus simple baseline visual question answering arxiv preprint
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chr grammar formalism first report arxiv jun henning christiansen roskilde university computer science roskilde denmark henning abstract grammars written constraint handling rules chr executed efficient robust parsers provide straightforward treatment ambiguity abduction integrity constraints well dynamic hypothesis generation techniques fit naturally grammars exemplified anaphora resolution coordination text interpretation introduction language constraint handling rules chr introduced tool writing constraint solvers traditional constraint domains real integer number arithmetic finite domains chr however turned general value available extension among others sicstus prolog chr web pages contains growing collection applications special interest language processing shown chr adds evaluation prolog flexibility combine topdown taken step showing abduction integrity constraints expressed straightforward way chr investigate chr applied grammatical formalism appears grammar written straightforward way propagation rules chr execute robust parser handles ambiguity without backtracking constructed naive way parsers high typically cubic worse means pragmas provided current implementations chr realistic almost linear parsers achieved executing quite similary parsers techniques easily expressed ambiguous grammars made unambiguous using simplification rules instead propagation rules similarly definite clause grammars dcgs chr grammars deliberately call take arbitrary attributes similar way accepts wider class bases except direct loops grammar chr grammars works correctly basis dcg evaluation backtracking may lead combinatorial explosion time avoided evaluation examines hypotheses space however significant differences importantly way arbitrary hypotheses produced consumed chr applied controlling parser evaluation semantics assumption grammars use linear intuitionistic assumptions various ways implemented easily chr abduction integrity constraints works fine well examples given anaphora resolution coordination text interpretation abduction related work notion constraints slightly different meanings often associated language processing constraint grammars unification grammars often used feature structure grammars constraint programming techniques applied complex constraints arise natural language processing see introduction overview constraint logic programming terms using constraint solvers applied many occasions well propose capture fundamental processes parsing semantic analysis means declaratively specified constraint solvers specifically expressed implemented language chr results promising still little work report context basic principle using chr propagation rules parsing also suggested elaborated used chr evaluation feature structures hpsg assumption grammars inspired linear logic provide structured way generate manage dynamically generated hypothesis provides elegant solutions problems otherwise difficult handle although seems need detailed methods control scope hypotheses provide new suggestion seems possible implement variety mechanisms chr based framework close similarity representation abduction assumption grammars chr seems indicate closer relation two still needs investigated references related abduction given section overview section explains propagation rules provide obviously correct parsers section concerns time complexity section shows assumption grammars implemented discusses improvements section introduces abduction integrity constraints model final section provides summary discusses possible syntactic sugar chr grammars perspectives introduction chr given refer online manuals chr grammar formalism basic principle set propagation rules chr constitutes natural evaluator set initial hypotheses given query evaluator produces answer set atoms logical consequences course provided infinite loop occurs set initial constraint store final constraint store rules apply applied using chr parsing represent string analyzed set initial constraints form token prolog atom indicates position string peter likes mary represented token token token nonterminals say grammar represented binary constraint symbol interpretation substring positions comprises instance grammar rule nonterminal terminal nonterminal represented propagation rule terminal token nonterminal empty productions handled inserting etc initial constraint store following consider grammars without empty productions may distinguish particular nonterminal grammar start symbol example grammar sentence verb peter mary verb likes translated following chr program verb sentence token peter token mary token likes verb parser given way grammar obviously correct following properties hold let input string start symbol grammar contains loops nonterminal derive parser guaranteed terminate given premise also nonterminal generate substring final constraint store contains constraint final constraint store contains derive grammar ambiguous substring corresponds different parse trees top node final constraint store contains exactly constraints argument list namely properties express usual notion correctness parser property means parser robust sense entire string conform grammar still possible extract parts represent recognizable subphrases property shows ambiguity handled natural way without backtracking constraints extended argument representing parse tree semantic representation possible trees meanings read final state properties desirable especially natural language processing language usually richer grammatical formalization ambiguity inherent property however property shows also final constraint store may become quite large contains nonterminals possible recognize even case contribute overall parse due local ambiguity using parser grammar string length result final state constraints extra arguments added nonterminals way dcg string indices hidden means syntactic sugar dcg final section discuss possible syntactic sugaring stay goal namely investigate full chr language grammatical formalism declarative semantics chr provides semantics grammars also refer procedural semantics implemented features various adjustments time complexity time complexity constraint solvers similar chr propagation rules studied shown parser grammar chomsky normal form max two grammar symbols rhs runs time grammar rules length input string conforms classical result parsing algorithm works quite similarly parser comprised propagation rules see background made empirical test analyzing random sequences propagation rule parser grammar grammar extreme degree local ambiguity resulting huge number derived constraints test sicstus prolog implementation chr shows indeed complexity exponential factor takes around probably due active garbage collector final store contains average constraints complexity increases higher powers grammar symbols appear rule believe interesting grammars produce far less final constraints still makes method unrealistic practical application fortunately several ways reduce time complexity illustrate parser arithmetic expressions based straightforward ambiguous grammar one rules following exp token exp exp current chr implementations recognize pattern recurrence variables test rule twice whenever new exp created check matches leftmost exp rule secondly rightmost addition entering token also result check applicability rule chr pragmas applied force parser work like classical parser making rightmost grammar symbol passive rule exp token exp exp pragma passive passive means whenever new exp constraint created one test generated rule namely potential match rightmost exp lhs rule advent token never trigger rule modified constraint solver produces exactly constraints original one even order parts removed computation process lot tests applicability rules anyhow destined fail made formal analysis time complexity modified parsers tests variety grammars indicate linear almost linear behaviour another source inefficiency potentially large number constraints pile state fewer final constraints also make easier interpret result parsing process one way reduce apply chr reduction rules instead propagation rules write instead effect constraints matched lhs rule removed constraint store thus fewer parse trees recognized thus reduction rules used parts grammar known unambiguous alternatively seen method make ambiguous grammar unambiguous mixture propagation reduction rules used degree ambiguity however handled reduction rule means backtracking using disjunctions rhs rule launching different hypothesis constraint store time since position tokens explicit rules also let parsing depend symbols sequence reduced one example incorporate based see following grammar arithmetic expressions use chr simpagation prove one needs argue terms fairly detailed model chr execution model rules constraints matched left backslash stay constraint store ones removed test front vertical bar guard assume text followed terminal symbol eof following rule says exp exp reduced provided next input token specified list token exp token exp member eof exp passive pragmas added described provided symbol lhs rule identified means indices rule token front backslash one thus one trigger rule together following rules pragmas understood grammar arithmetic expressions traditional associativity precedence token exp token exp member eof exp token exp token exp exp token exp token exp token int integer int exp result achieve efficient parser constraint store used effectively stack parser performs exactly steps comparisons traditional parser conclude time complexity problem chr parsers flexibility available grammar writer consider symbols terminals left right symbols reduced combine chr different kinds rules pragmas different effects add new tokens rhs shown etc addition parser may refer constraints represent purely syntactic hypothesis appears following assumption grammars beyond chr possible let rules produce consume arbitrary hypotheses consider following sketch two chr grammar rules first rule produces hypothesis say extracted context rule applied second rule executed hypothesis present value becomes available anaphora natural language treated passing hypotheses constraint store way assumption grammars include specialized operators managing hypotheses show implemented applied chr grammars assumption grammar expression asserts linear hypothesis used following text means expression called expectation asserting hypothesis means used represent assertion constraint predicate symbol argument list split technical reasons indicates position string hypothesis supposed created two operators represented analogous ways deviating slightly syntax achieve symmetric notation introduce three new operators hypotheses whose meanings similar except hypotheses used consumed order leave string index following simplification simpagation rules provide implementation chr true true true true true true explicit unification guard serves test unifiability well porting argument value asserted hypothesis supposed ground correct derivation assumption grammar requires expectations matched corresponding assertions end optional test available meaning linear hypotheses consumed similar facilities easily implemented means chr auxiliary however use hypotheses introduces one technical problem chr give parse tree specific set hypotheses thus ambiguous grammar propagation rules would mix hypotheses corresponding different trees present assume unambiguous grammars use simplification rules give excerpts sample grammar inspired demonstrating anaphora coordination occurrence proper name individual following rule introduces hypothesis individual referred subsequently pronoun gender gender gender pronoun gender gender gender coordination arise relation ellipses mary likes martha hates peter object first sentence implicit shared second sentence simple full sentences described follows chr matches constraints processed common instance condition rule true would apply fewer cases corresponding one sub verb obj sent sub obj following two rules take care coordination notice ellipsis identified second two sentences offers object anyone missing object token sub verb obj sentence sub obj sent token sent sent obvious rules analysis mary likes peter loves martha hates yields following semantic representation mary peter mary peter martha peter however example gives rise discussion assumption grammars first sentence preceding happens complete asserted hypothesis used may interfere analysis subsequent ellipses avoid chr rule refined asserts hypothesis need one replacing unconditional assertion following piece code true general believe detailed control scope asserted hypothesis needed feasible assertion grammars explicit string indices useful well primitives chr seems suggestions scoping mechanism another feature missing assumption grammars selection best hypothesis say resolving pronoun considering distances string hypothesis applied often sketch rule follows constraint backslash serves test applicable hypotheses gender pronoun gender select best hypothesis form gender according criteria remove constraint store gender hypotheses feasible tested backtracking leave competent linguists design mechanisms indicated chr seems provide framework implementing scoping provided implications goals seem sufficient hypotheses may relevant say position others addition hypotheses may temporarily overruled others abduction integrity constraints several authors noticed abduction useful way conceive implement aspects natural language interpretation deductive approach exercised section represents meaning text complex structure attached start symbol synthesized meanings constituents abduction meaning appear assumptions generated parallel syntactic analysis needed integrity constraints check dynamically collected set hypothesis consistent consider following dcg rule instance nonterminal understood presence good phrase true sentence noun phrase referring existing object related instance precondition composition good subphrases good phrase formulating language interpretation abduction unknown body assumptions entails set instances applied literature rich abduction procedures job shown abduction integrity constraints expressed chr following illustrate integrate chr parser however without caring formal relation dcg rule written following chr grammar rule applying introduce constraint store instance necessary previous dcg rule hypotheses generated example section seen part abductive explanation text could sensibly correctly uttered gender individual may recognized proper name lexicon deduced context thus following integrity constraint relevant masc fem fail sentence meanings removed nonterminal instead launched constraint store sub verb obj fact sub obj sent different anaphoric resolutions thinned along way means semantic integrity constraints fact likes fact hates fail fact loves fact hates fail fact hates fail likely produce several instances hypothesis chr means avoid provide correct interpretation tricky combinations mary likes martha hates abduction provides also way work negative hypotheses means explicit negation expressed sketch integrity constraint fail means anything asserted opposite proved indicated chr grammar formalism seems capable integrating abductive text interpretation several benefits explicit synthesis meanings however principle illustrated still needs refined formal grounds established described grammar rules represented simplification rules otherwise different hypothesis sets different parse trees mixed thus alternatives explored backtracking rule bodies however seems matter proper engineering find ways manage sets alternative mutually exclusive hypotheses constraint store weighted abduction seems fit model structured approach based fuzzy logic could used integrity constraints also behave less rude way simple wrong hypotheses vanish instead provoking failure thus backtracking conclusion perspectives called chr grammars relates chr way dcgs relate prolog grammar rules written systematic way rules host language used directly parser inherits computational characteristics underlying machinery dcgs work backtracking chr grammars works addition provide evaluation strategy better suited parsing especially natural language chr paradigm supports use dynamically generated hypotheses abduction assumption grammars extremely useful natural language analysis experiences reported indeed promising use chr grammatical formalism provides high degree flexibility expressive power language processing believe exposed small portion comparison dcgs may suggest add layer syntactic sugar suppress arguments syntactic indices introduce devices replace different uses take simplification rule arithmetic plus shown section means suitable preprocessor may suggest write following nice way exp exp eof exp prefer keep order lhs rhs rules indicate bottomup nature let choice propagation simplification rule visible using symbols dcg curly brackets used indicated code bypassed preprocessor notice relevant apply sides chr grammar rule marker indicates reference right context sequence reduced analogous marker used beginning rule indicate references left context add passive pragmas leftmost grammar symbol lhs described section rule may prefixed operator rulelr global directive affect rules mode forced may affect final result chr grammars using reduction rules investigation also exposed interesting topics future work study possible relation abduction assumption grammars perhaps instances general framework hypothetical reasoning design facilities controlling scope hypotheses pointed section technical work needed order achieve full version abduction plans near future include extraction noun phrases used danish search systems developed parallel implementation using traditional language processing tools grammar substantial subset danish consideration contextual perhaps kind semantic analysis acknowledgment research supported part ontoquery funded danish research councils copenhagen references abdennadher operational semantics confluence constraint propagation rules third international conference principles practice constraint programming lncs november abdennadher christiansen experimental clp platform integrity constraints abduction proceedings flexible query answering systems advances soft computing series springer abdennadher flexible query language flexible query answering systems lnai allen natural language understanding benjamin cummings second edition aho ullman theory parsing translation compiling volume parsing aho sethi ullman compilers principles techniques tools charniak mcdermott introduction artificial intelligence addisonwesley constraint handling rules online http christiansen declarative semantics language bruynooghe proc second workshop logic april leuven belgium christiansen automated reasoning metainterpreter journal logic programming special issue constraint logic programming christiansen open theories abduction context accommodation international interdisciplinary conference modeling using context context bouquet brezillon serafini eds lecture notes artificial intelligence dahl tarau assumption grammars processing natural language proc fourteenth international conference logic programming mit press decker extension sld abduction integrity maintenance view updating deductive databases proc jicslp pages denecker schreye sldnfa abductive procedure abductive logic programs journal logic programming duchier constraint programming natural language processing lecture notes esslli august constraint handling rules lecture notes computer science theory practice constraint handling rules journal logic programming vol gabbay kempson pitt labeled abduction relevance reasoning nonstandard queries nonstandard answers demolombe imielinski eds oxford science publications gabbay reyle extension prolog hypothetical implications journal logic programming vol hobbs stickel appelt martin interpretation abduction artificial intelligence kakas kowalski toni role abduction logic programming handbook logic artificial intelligence logic programming vol gabbay hogger robinson eds oxford university press mcallester theorems talk abstract proc principles practice constraint programming international conference dechter lecture notes computer science meyer framework diagrammatical reasoning journal applied artificial intelligence special issue constraint handling rules miller lexical scoping universal quantification proc sixth international conference logic programming eds levi martelli mit press penn applying constraint handling rules hpsg proc first workshop constraint reasoning programming workshop first international conference computational logic july imperial college london available electronically http corr http pereira warren definite clause grammars language analysis survey formalism comparison augmented transition grammars artificial intelligence sicstus prolog user manual version sics swedish institute computer science http
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mar ensuring referential integrity causal marc shapiro annette bieniusa sorbonne paris france inria paris france kaiserslautern germany peter zeller gustavo petri kaiserslautern germany irif paris diderot paris france abstract referential integrity important correctness property shared distributed object storage system sometimes thought enforcing requires strong form consistency paper argue causal consistency maintain support argument pseudocode reference crdt data type maintains causal consistency quickcheck found errors model purpose paper construct reference data type demonstrating causal consistency progress guarantees ensure implement safe deletion operation support claim pseudocode solution sketch paper uses form reference counting designed distributed systems called reference listing objects reference list must deleted acm reference format marc shapiro annette bieniusa peter zeller gustavo petri ensuring referential integrity causal consistency papoc referential shop principles practice consistency distributed data april consistency porto portugal acm new york usa pages https references referential integrity consider shared store memory objects reference data type linking objects store let call referencing object source reference referenced object target intuitively referential integrity invariant states application reference target target exists sense application access target safely referenced object must deleted conversely object reached reference deleting allowed say object unreachable target reference never future latter clause problematic weak consistency property wish achieve following safety object deleted unreachable liveness unreachability object eventually detected storage system application delete objects explicitly programmer must careful preserve invariant problem studied context concurrent garbage collection decades folklorically often thought enforcing requires synchronisation strong consistency fact previous work stated otherwise main produces permission block copyright information acm acknowledges contribution authored employee contractor national government government retains nonexclusive right publish reproduce article allow others government purposes papoc april porto portugal association computing machinery acm isbn https integrity causal safety property instance implication invariant reference object exists object accessed deallocated elementary logic tells sequential pattern making true followed making true maintain invariant backward pattern similarly making false followed making false forward pattern also works backward pattern translates allocate object assign reference forward pattern delete references object delete concurrent system causal consistency two updates ordered processes observe order therefore expect patterns extend system unfortunately maintain patterns may executing parallel encouraging remember datatypes engineered support concurrent updates instance set support concurrent insertion removal element making one operation win deterministically one superseded however design directly since handling references also requires handle referred objects accordingly including transitive reachability furthermore easy ensure safety never deleting anything also require liveness note causal consistency safety property allows arbitrarily old versions observed need add progress guarantee assumption ensure algorithm live assume objects interest accessed via reference datatype discussed herein address complex problem objects accessible via external means key url via database query called root objects garbagecollection parlance purposes never deleted papoc april porto portugal marc shapiro annette bieniusa peter zeller gustavo petri object object object object null object object null null initial state object object object replica replica figure concurrently creating references deleting objects lead dangling references replicas reconciled object outref invoke target makes sense single target outref contains multiple values invocation fails application performing new assignment figure illustrates three source objects containing attribute single outref named respectively two target objects state illustrated might result following code snippet init object object inref init init algorithm design hinges two principles implemented assuming causal consistency outref assigned source object initialisation assignment object ensure corresponding inref added target init object object importantly causal consistency enough enforce inref ordering updates delete target require ref exists later added target property checked mechanisms rely causal consistency progress guarantees combination figure references concurrently asof properties ensure introsigning twice duction description sketch following reference handling protocol pseudocode description given appendix source object contains instance data type called outref every attribute refers another object target object associated exactly one inref inref sources pointing target creating new reference initialises inref outref operations supported inref initialisation testing whether deleting target allowed source object contains number distinct outrefs outref supports following operations initialisation assigning another outref iii assigning null assume deleting object automatically nulls outrefs invocation detailed shortly support concurrency assigning outref behaves much like register assignment overwrites previous value concurrent assignments occur resulting reconciled value contains values simplify semantics check side assignment system model pseudocode pseudocode references listed appendix preliminary explanations required references layered unmanaged addressing mechanism similar memory address used jvm call key key uniquely single discrete possibly replicated object system model based invocation split two phases generator executes single replica generates list downstream messages eventually received replicas executed corresponding source replica downstream messages processed atomically generator replicas may observe delays different downstream messages always receive order generator generator may check preconditions noted precond shared state precondition false operation fails generator may side shared state must every replica therefore may depend testing shared state assume operation preconditions stable evaluating ensuring referential integrity causal consistency papoc april porto portugal precondition true change concurrent operation assume causal consistency one operation delivered replica operations visible consider two alternatives composed operations atomic operation atomic composition resp compose single atomic generator resp somewhat similar transactions without isolation property pure causal updates single object chained respecting order code somewhat similar transaction chaining cases precondition false whole operation fails appendix provides pseudocode latter logic relatively simple creating copying reference avoid races following backward direction adding target source resetting removing reference follow forward direction removing source target deal concurrency ensuring every reference unique careful losing information details tedious hopefully explained comments operation merits detailed explanation operation returns true inref argument reachable however order break circular reference patterns argument lists references ignore stably notation third invariant means assertion true concurrent mutations could make detecting stably boils detecting termination implementation well understood requiring replicas know order exchange information progress note causal consistency usually safety guarantee order ensure stably check eventually succeeds must add assumption progress reads return old version earlier drafts implementation able updated implementation tests problem random executions acknowledgments research supported part european project number lightkone rainbowfs project agence nationale recherche france number references mustaque ahamad gil neiger james burns prince kohli phillip hutto causal memory implementation programming distributed computing march https peter bailis alan fekete michael franklin ali ghodsi joseph hellerstein ion stoica coordination avoidance database systems proc vldb endow https int conf large data bases vldb waikoloa hawai usa koen claessen john hughes quickcheck lightweight tool random testing haskell programs proceedings fifth acm sigplan international conference functional programming icfp https paulo ferreira marc shapiro garbage collection dsm consistency symp sys design implementation osdi acm monterey usa http paulo ferreira marc shapiro larchant persistence reachability distributed shared memory garbage collection int conf distributed comp sys icdcs hong kong https alexey gotsman hongseok yang carla ferreira mahsa najafzadeh marc shapiro cause strong enough reasoning consistency choices distributed systems symp principles prog lang popl petersburg usa https leslie lamport time clocks ordering events distributed system commun acm july http cheng daniel porto allen clement johannes gehrke nuno rodrigo rodrigues making systems fast possible consistent necessary symp sys design implementation osdi hollywood usa tobias nipkow lawrence paulson markus wenzel proof assistant logic lncs vol springer marc shapiro peter dickman david ssp chains robust distributed references supporting acyclic garbage collection rapport recherche institut national recherche informatique automatique inria rocquencourt france http marc shapiro nuno carlos baquero marek zawirski replicated data types int symp stabilization safety security dist sys sss lecture notes comp xavier franck petit villain eds vol grenoble france https marcin skubiszewski patrick valduriez using cuts concurrent garbage collection networking information systems paolo viotti marko consistency nontransactional distributed storage systems jul https gene wuu arthur bernstein solutions replicated log dictionary problems symp principles dist comp podc vancouver canada http correctness order validate correctness crdt references implementation formalized system model pseudocode implementation tested haskell quickcheck corresponding code available quickcheck tests generate random executions check third invariant described pseudocode generate interesting random executions let generated event depend two randomly chosen previous events randomly decide many messages delivered new event likely event observes events partially common source bugs indeed able discover pseudocode pseudocode also makes use need stable use term stable section follows terminology used relyguarantee logic atomic version easier read prefer minimise assumptions obtained version replacing chained single atomic one text called stable property literature distributed algorithms use stably distinguish usage footnote https following pseudocode describes version references atomic version essentially replacing cascaded single atomic one block structure indicated indentation comments preceded character datatype outref papoc april porto portugal reference object type containing inref marc shapiro annette bieniusa peter zeller gustavo petri key embedding object object multiple outrefs assume one distinct register key routing information referenced object outkey key uid initial nullkey nulluid datatype inref key embedding object register key set reverse references inkey key uid initial emptyset call init inuse initial false correct type invariant forall outref create reference outref inref init outref outref inref inref generator outref inref precond inref inuse call init effector inref run effectors effector outref effector inref true remove inref deleting object embeds inref calls therefore outer delete fail remaining references reset inref inref generator outref exists inref exists monotonic invariant precond inref forall outref effector inref skip finally remove old reverse refs inref remove remove outgoing reference deleting object embeds outref calls reset outref outref generator outref assigning nullkey outref nullkey inref unreachable remains unreachable invariant forall inref outto outref outval outref stably true replicas concurrent updates flight assign outto outval stably emptyset henceforth emptyset outto outval constructor part api copy outval outto reset outto order either may key inref inref initially null outval target already outto inref embedded inside object key inref concurrent assign outref store multiple values inside user resolve subsequent assign constructor part api key outref outref assign outto outref outval outref outref embedded inside object key generator outto outval outref simplification ensure outval one target local check necessarily stable updates outref new key value part api outval count outto outref key let outval generator outto outto newkey let outto let outto getall use reference call target object let newuid deref outref outref invocation invocation explicit effector chaining generator outref invocation nullkey local checks necessarily stable effector outto newuid outref count else outref nullkey effector outto newuid let outref effector invocation effector outto newuid invoke invocation first insert new target inref add newuid target object reliably referenced effector outto newuid tested generator exist still effector reset shortly effector outto newuid inref inref assign source set outref default emptyset outto newuid conc assign possible boolean forall generator inref chain reset check remaining effector none default effector set uid ensuring referential integrity causal consistency fold lambda cons return stably inref effector skip papoc april porto portugal figure available jpg format http
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value distributed energy resources heterogeneous residential consumers siddharth patel ram rajagopal oct department civil environmental engineering stanford university department electrical engineering courtesy stanford university abstract presence rooftop storage residential sector poised increase significantly quantify detail value technologies consumers service providers characterize heterogeneity household electricity cost savings prices due consumption behavior differences top consumers benefit two three times much remaining different pricing policies significantly alter households fare respect one another define value information financial value improved forecasting capabilities household typical value information cents per hour per kwh reduction standard deviation forecast error coordination services combine resources available households reduce costs additional original total cost surprisingly basis coordinated action alone service providers encourage adoption beyond within group coordinated information however enables providers generate additional value increasing adoption distributed energy resources ders essential part modernizing grid pose challenges design management operation electricity system dramatic changes expected consumers adopt ders become prosumers capable responding prices signals grid operators technology vendors solar city der resource aggregators ohmconnect play significant role emerging ecosystem making ders accessible smaller consumers ensuring operate technologies manner aligned compatible needs grid system proliferation ders create opportunities new business models well coming impact adoption energy storage rooftop photovoltaic systems residential consumers well understood impact ders depends adoption rate technology operations strategy financial policy arrangements system participants consumer behavior resulting consumption pattern heterogeneity critical drivers der value govern interactions dimensions residential electricity consumption exhibits significantly diversity previously believed yet heterogeneity seldom accounted existing studies use data small number residential commercial consumers evaluate economics storage combined paper propose simple scalable methodology estimate der value impact incorporates consumption heterogeneity apply methodology provide first kind assessment value technologies residential consumers technology vendors aggregators various business models policy arrangements focus batteries commercially available gaining traction costs decrease study accounts adoption rates consumption heterogeneity unique ways identify tipping points impact utilize large dataset hourly power consumption recordings residential consumers northern california capture heterogeneity based data build various models assess value storage consumers form bill savings estimate value delivered entities provide improved information enable resource sharing among residential consumers households rely information future consumption generation deciding operate ders model constraints information impact value ders households analysis gives estimate value services improve accuracy information available households communication technology enables aggregators coordinate share ders among group households analyze value coordination services generate depends pattern technology adoption households sharing mechanism assessing value data storage devices enable households shift energy usage time rooftop generates electricity households consume directly sell back grid technologies allow households reduce electricity costs estimate much households groups households could save bills adopted rooftop system kwh storage operated devices minimize electricity costs take household hourly smart meter data inflexible consumption provide snapshot potential bill savings thus electricity prices exogenous model similarly assume households significantly alter electricity consumption behavior adopt ders face differing rates refer methods section detailed explanations data sources analyses value technology households define absolute savings household much electricity bill decreases adopts storage household normalized savings absolute annual savings divided original total energy consumption gives cents per savings estimate household relative savings ratio household normalized savings top household given policy max relative savings always enable comparison concentration benefits different pricing policies compute savings three pricing policies incorporating tou wholesale dynamic rates listed table figure conducted five sensitivity studies varying storage device sizes qualitative results study unchanged count count policy purchase price retail tou retail dynamic retail tou sale price wholesale discounted dynamic discounted tou peak hour peak fraction price coordinator mean load price hour figure considerable heterogeneity households dataset together consume gwh electrical energy annually fig shows distribution mean electrical load households dataset range fig shows distribution fraction consumption takes place peak hours range study household given inflexible load kwh storage device rooftop system connected house bus inverters table defines pricing policies used simulations retail dynamic rate exposes households directly wholesale market price variations fig plots retail tou rate summer red winter blue business day figure wholesale rate retail dynamic rate scaled version wholesale rate heterogeneity savings figure illustrates distribution annual absolute savings different pricing policies variation annual savings households greatest policy case sale price much less purchase price benefit offsetting consumption much greater selling electricity back grid thus heterogeneity household consumption patterns drives greater differences savings contrast policies sale price close purchase price offsetting consumption little better selling electricity back grid thus variation household savings reduced given size battery policy impact savings policy fundamental impact absolute savings available households policies advantageous allowing households sell back electricity almost full retail rate switching policy reduces sale price wholesale rate cuts available savings half compared policy households fare better policy policy dynamic rate effect pronounced version tou higher peaks lower troughs thus policy worth higher peak hour prices coincide peak generation hours storage devices also higher price differentials work upon segmentation households normalized savings figure shows three pricing policies households divided three segments based normalized savings top middle bottom average normalized savings among top savers twice rest policy thrice rest policies concentration benefits significant implication vendors could provide rooftop storage households stand save top savers represent pool greater potential surplus first vendor enter market would reap substantial benefits targeting later entrants would compete remaining households relatively similar lower potential savings policy impact household savings rank figures compare savings different policies absolute savings ranking households similar two policies sale price close purchase price policies somewhat different sale price much lower purchase price policy said correlation ordering strong across three policies ranking households normalized savings almost identical three pricing policies thus pricing policy strong impact magnitude savings weak effect ordering households best relative savings savings rank rank figure absolute annual savings plotted policy red policy blue policy green pricing policy households ranked decreasing order based savings half available savings policy eliminated switching policy policy savings halfway two relative savings plotted three pricing policies top saver segment labeled middle bottom policy mean savings among top households times mean savings rest ratio policies concentration presents opportunity targeting absolute annual savings different policies plotted spearman rank correlation coefficient given plot ordering households absolute annual savings similar policies ordering policy less strongly correlated normalized savings different policies plotted household ordering practically three policies shown absolute annual savings negatively correlated normalized savings strongly policy weakly policies value information deciding operate storage device household relies forecast future consumption rooftop generation preceding sections assumed households perfect foresight quantities imperfect forecast leads household operate storage suboptimally saves less money would forecast perfect thus characterize value information household evaluating much pays electricity forecast error level increases metric gives sense much household willing pay service provider reducing forecast error data analytics improved algorithms household increase coefficient forecast error perfect foresight compute much cost electricity increases absolute normalized terms given policy pricing relationship cost highly linear range thus slope best fit line cost provides good estimate sensitivity household electricity cost forecast error sensitivity define value information given household heterogeneous consumption behavior leads variation value information households figure shows distribution value information absolute terms households annual value information greater per unit words assuming adopt storage households willing pay annually service reduces standard deviation forecast errors amount equivalent one third mean consumption distribution value information normalized terms shown figure mode around standard deviation forecast error households willing pay hourly reducing standard deviation forecast errors kwh insignificant compared purchase price electricity policy considering households consume average less kwh hour value coordination define coordinator entity collectively manages storage rooftop group households coordinators aggregators provide services consumers enabling sharing existing assets also provide services grid ecosystem limiting ramping events caused synchronized activity shaving peak load regulating voltage frequency focus additional value coordination bring group households includes adopters define value coordination additional savings coordinator achieve group households beyond achieve acting separately additional savings represent fund coordinator draw upon compensate adopters allowing coordinator use devices benefit others group value coordination sum value coordinated action vca value coordinated information vci coefficient variation defined standard deviation forecast residuals divided mean actuals density cdf value info value info figure value information increase electricity costs incurred household due increase forecast errors fig shows cumulative distribution function value information absolute terms fig gives distribution value information normalized terms blue line fitted density deviation values policy pricing value coordination depends adoption rate pattern pattern households adopt technology subject complex economic behavioral social factors consider two simple adoption rules forward adoption adopters stand save absolute annual savings terms reverse adoption adopters stand save least also consider random adoption value coordinated action coordinator use capacity storage device one home order shift energy usage homes coordinator also ensure electricity generated rooftop homes group used offset consumption within group rather exported grid essence coordinator manages group households one large aggregated household large system storage device redirecting technology capacity uses decrease group cost define vca additional savings coordinator achieves group households beyond save act separately perfect foresight case vca substantial original total cost electricity shown figure vca increasing adoption rate point three adoption patterns coordination provides increasing value increasing adoption point beyond adoption value coordination decreases reason decline follows tou rate expensive peak period inexpensive period acting alone minimize cost electricity household use devices offset consumption peak period first spare capacity offset consumption times sell electricity back grid coordinator hand minimizing entire group cost electricity therefore use capacity available first offset much peak period consumption across households possible thus coordination adopters home energy technology capacity goes offsetting peak period consumption means greater savings reason vca much lower policies sale price close purchase price level adoption low many households technology therefore need help offsetting peak period consumption coordinator provides help level adoption high households technology offset peak period consumption coordinator fewer opportunities redirect technology capacity offset peak period consumption instead ends offsetting period consumption selling surplus electricity back grid accrue less savings thus adoption increases beyond certain point coordinator left opportunities redirecting capacity setting basis vca alone coordinator would encourage der adoption beyond group households would reduce value services nonetheless even adoption coordinated action value households consume enough electricity make good use households storage addition mitigating inefficient adoption coordination captures value would otherwise lost due inefficient adoption patterns ensuring technology capacity redirected efficient application example reverse adoption households stand save least adopt first coordinator well redirecting capacity group save thus vca higher reverse adoption forward adoption even adopters highest total annual savings may take full advantage der capacity every day vacation forward adoption coordinator still able realize additional value reallocating underutilized capacity high adoption rates vca forward adoption remains high fraction maximum later adopters save least ders redirecting capacity elsewhere remains valuable service value coordinated information operating aggregated storage devices adopters coordinator needs forecast aggregate quantities thus faces much lower forecast errors individual households enabling achieve optimal outcome define value coordinated information additional savings beyond vca coordinator achieves presence forecast error vci increases adoption level forecast error adoption patterns shown figure adoption pattern much effect vci higher levels adoption vci increases given increase forecast error level adoption higher coordinator operating greater amount technology capacity forecasting advantage yields greater amount additional savings figure shows value coordination sum vci vca forward reverse adoption patterns forecast error level case forward adoption higher levels adoption increasing vci offsets drop vca coordinator would encourage adoption beyond general value coordination may increasing decreasing adoption rate depending adoption pattern adoption level forecast error level vca adoption figure value coordinated action vca given percentage baseline total cost electricity households prior technology adoption purple curve forward adoption pattern orange curve reverse adoption gray curve random adoption vca increases level storage adoption within group increases coordination provides less value vca greater reverse adoption pattern coordinator able redirect capacity therefore overcome inefficient initial allocations technology values computed policy pricing vci vci vci adoption adoption forward adoption adoption random adoption reverse adoption vca tcs tcc value coordination vci value coordination figure plots show value coordinated information vci function adoption level forecast error level forward random reverse adoption patterns graph lightest line corresponds forecast error darkest intermediate colors increments vci greater forecast error greater adoption level greater vci reported percent baseline total electricity cost households fig values policy pricing adoption adoption forward adoption reverse adoption figure diagram illustrates definitions vca vci tcs total cost electricity paid group households optimize separately tcc total cost pay coordination light gray bars costs perfect foresight dark gray bars additional costs due forecast error vca reduction cost perfect foresight vci additional reduction cost presence forecast errors sum vca vci value coordination value coordination forecast error plotted forward reverse adoption patterns lighter color vca darker color vci fig values reported percent baseline total cost electricity without technology policy prices apply discussion methodology results useful stakeholders developing plans action respect ders provide tool policy makers understand impact different pricing policies household incentives ordering pricing policy useful lever adjust amount savings available households hence economic incentives adopt ders however significantly alter households fare best adopting ders purpose policy tools targeted subsidies would required enabling sharing arrangements allow coordination another way policy makers ensure many households possible benefit der adoption coordination even households consume electricity ones adopt ders households share capacity reduce costs well end households better coordination technology capacity deployed efficient applications study reveals key insights emerging business models serve households adopt ders magnitude distribution household savings give sense market size structure vendors der equipment vendors best households largest potential savings would well support pricing policies generous households like policies two policies also result even benefits across households would make targeting less priority vendors sales operations take place absolute dollar basis hand policy targeting important absolute savings rather unevenly distributed contrast vendors operate normalized per kwh basis must prioritize targeting matter market segmented similarly pricing policies normalized savings negatively correlated absolute savings meaning top market segment composed households save large amount total whose consumption behavior aligned well incentives created pricing policies finding underscores importance devising arrangements business models coordinating der capacity among groups households many would adopt could collectively benefit great deal shared resource value information analysis gives clear guidance information service providers potential price points algorithms analytics help households manage ders similarly value coordination analysis gives aggregators sense revenue possibilities sharing ders among households also identify tipping point aggregators adoption rate within group households exceeds certain level vca declines presence uncertainty however aggregator deliver vci increases adoption rate combined value coordination may increase decrease increasing adoption depending adoption pattern adoption level uncertainty level may case aggregator delivers maximum value assembles households groups include adopters particular mix finally note tension business models equipment vendor coordination service provider equipment vendor interested selling storage systems households best targeting highest potential savings pay equipment inclined proceed along forward adoption pattern charge higher prices get customers hand coordination service provider would want encourage reverse adoption pattern makes services valuable however coordination service provider encourage adoption beyond point value services maximized adoption figure tension two business models could manifest support competing policies dealing pricing equipment subsidies sharing mechanisms methods study incorporates actual smart meter data pricing data solar irradiance data specifications currently available home energy technologies assume households groups households operate technologies minimize cost electricity describe detail data sources use analyses perform data sources household consumption household electricity consumption data comes residential smart meters one year period spanning august july households customers pacific gas electric company california meters include single family homes apartments meters low consumption annual mean excluded meters high amount zero readings readings treat smart meter data household inflexible load move rests two assumptions first consumption behavior change due rate changes thus even though almost households inclining block rate period time meter data electricity consumption behavior change exposed pricing policies study note retail rate design typically aims revenue neutrality means different rates marginal price faced consumers varies much average price find household electricity consumption responds strongly average price electricity lending support using meter data inflexible load second assumption consumption behavior change due der adoption sizes systems study suggests households would increase overall electricity consumption computing bill savings based greater consumption would result larger savings report study conservative sense prices retail tou utilities moving towards rates create incentives consumers shift electricity purchases periods rate lower option serves retail tou rate study table contains relevant elements tariff exclude fixed charges tariff computing household electricity bills discounted tou recent survey industry professionals asked households receive electricity sell back grid plurality favored full retail rate minus costs using physical infrastructure grid costs estimated retail rate households would compensated retail rate call discounted tou june september october may peak hours kwh kwh peak hours kwh kwh table retail tou rate study comes peak hour rates apply weekdays holidays weekends hours charged peak rate wholesale locational marginal prices lmps serve wholesale price come wholesale energy market administered california independent system operator caiso use day ahead lmps published caiso dates corresponding household consumption data set negative prices zero wholesale rate varies location use rough mapping household zip code nearest caiso pricing node latitude longitude use lmp node wholesale price household thus households given zip code face wholesale price retail dynamic retail dynamic rate scaled version wholesale rate scaling designed maintain revenue neutrality retail tou day across zip codes total cost households inflexible load retail tou rate wholesale rate computed lmps zip code scaled ratio retail tou revenue wholesale revenue retail dynamic rate exposes households directly fluctuations wholesale market discounted dynamic discounted dynamic rate dynamic rate home technologies assume households install ders behind meter without interference electricity provider households living apartments adopt ders participating building neighborhood sharing arrangement rooftop size rooftop system output pvusa test condition ptc equates array area according solar energy industries association average size installed residential systems united states years ago storage device storage device specifications based first generation kwh tesla powerwall capacity kwh maximum sustained charging rate also maximum sustained discharging rate round trip efficiency take square root split charging discharging efficiency assume interconnection assume rooftop connected house bus inverter efficiency storage device connected way thus energy flows storage device incur loss inverters assume limit power household draw provide grid solar irradiance class weather stations solar irradiance data california station fit beta distribution global horizontal irradiance ghi based statistics published national solar radiation data base nsdrb geographic information households zip code weather stations latitude longitude use rough mapping match zip code nearest weather sample beta distribution station generate solar irradiance households zip code households zip code get irradiance sample pricing scenarios analyze three different pricing policies listed table figure main text policy households purchase electricity retail tou rate sell surplus back grid wholesale rate utilities california proposed price structure recent discussions regulators generally considered least generous compensation scheme households likely encounter policy households purchase retail dynamic rate sell surplus back discounted dynamic rate policy households purchase electricity retail tou sell back grid discounted retail tou retail tou discounted retail tou rate households thus households face prices policy three scenarios electricity prices dynamic market clearing study words households price takers actions influence prices face operation storage device consider rooftop uncontrollable system produce however much power dictated solar irradiance storage device however controllable households determine operate study households solve linear program daily basis choose optimal schedule charging discharging battery describe linear program ith household hourly smart meter data year data jth day take household inflexible electrical energy consumption day multiply day solar irradiance household panel power rating get energy generated household system household system finally get household net load day san francisco international airport sfo weather station particularly low irradiance compared nearby areas therefore manually assigned zip codes near sfo weather stations area using average ghi reported nsrdb data viewer guide conventional residential retail rates change rapidly intents purposes retail tou rate study would fixed households begin uptake ders inverter efficiency section assume households perfect foresight coming hours let vectors hourly prices household buys sells electricity respectively given pricing policy let denote prices jth day day household must select sequence hourly actions storage device let denote action hour day positive device charging hour negative device discharging similarly denotes state charge device state charge end hour let charging discharging efficiencies respectively finally denotes hourly net exchange grid positive household drawing power grid hour negative household supplying power grid household seeks minimize pays electricity day solves following linear program minimize subject respectively maximum charge discharge rates storage device maximum capacity storage device state charge device end prior day operators represent application max min respectively let optimal schedule denoted household operates battery per minimized objective pays electricity day denote minimized objective value household savings analysis given pricing policy first compute baseline annual bill household absence home energy technologies bbl lti next compute household new bill presence technologies given pricing policy absolute annual savings bbl distribution absolute annual savings different pricing policies plotted figure normalized savings convert normalized savings relative savings dividing maximum value households given policy max figure plots distribution relative savings different pricing policies compute error bars absolute annual savings normalized savings bootstrap resampling days year household error bars small omit graphs value information analysis estimate value information additional cost incurred household due increase forecasting errors given day instead perfect foresight inflexible consumption generation household must generate forecasts note household know relevant prices without uncertainty retail tou change daily basis day ahead lmps published prior start day apply introduce forecast error based coefficient variation let independently identically distributed across forecast error normally distributed mean entries words mean hourly inflexible load ith household similarly normally distributed mean hourly solar generation household household uses forecasts schedule operation storage device let household solves linear program except replaces constraint optimal schedule solution household uses operate storage device household cost electricity case objective linear program true consumption solar generation different forecasts let true net load household cost electricity day household annual cost electricity level forecast error compute perform linear regression value information household absolute terms slope coefficient regression units cumulative distribution metric policy prices given figure get value information household normalized terms linearly regress multiplying value information absolute terms units standard deviation forecast error distribution metric policy pricing given figure multiplying total annual cost gives total annual cost dividing figure total annual load yields cost per unit load coefficient variation defined standard deviation forecast errors divided mean actual signal hourly load units standard deviation kwh standard deviation hourly kwh load dividing kwhc load kwhkwh yields units standard deviation forecast error hourly solar generation units still work standard deviation units kwhkwh cancellations clean case load choose report metric standard deviation forecast error solar generation scales directly load method value coordination analysis given scenario compute value coordination additional savings entire group households could obtain acted collectively instead individuals break value coordination two components value coordinated action vca value coordinated information vci total costs without coordination index households total number households given pricing policy let ordering position household households sorted decreasing order let ordering position household households sorted increasing order finally let ordering position household random permutation thus household saved policy wherever random permutation consider given adoption rate define afwd arev ardm define bfwd afwd define brev brdm analogously define total cost forward adoption tfwd bbl words add new bills technology adopters baseline bills without technology total costs reverse adoption randomp adoption defined analogously next define baseline total cost without technology tbl bbl figure plots tfwd trev trdm percent tbl policy pricing value coordinated action group minimizes electricity costs daily basis given day set optimization follows note vca perfect foresight take mean prices across households prices coordinator faces take given level adoption assume forward adoption pattern solar energy generated group generated systems adopters group inflexible load total across households group net load obtain group storage device capacity charging discharging rates scaling parameters household storage device number adopters coordinator solves following optimization problem schedule operation collective storage total cost baseline adoption figure total annual cost electricity without coordination households percent baseline total cost without technology tbl plotted percent households adopted storage analysis policy prices purple curve forward adoption pattern orange curve reverse adoption gray curve random adoption tbl million serves baseline cost used figs minimize subject coordinator controls adopters storage devices execute optimal schedule aggregate level generally many ways distribute adopters devices implicit assumption model coordination group households free use wires homes losses sharing energy way energy supplied grid one home group whether rooftop storage device used offset consumption separate home group loss fees losses induced first home inverter storage device inefficiencies effect model coordination treats group homes one big household combining consumption rooftop generation storage devices cost electricity group households day optimal value objective call define vca forward adoption pattern adoption level vcafwd tfwd cfwd vca reverse random adoption patterns defined analogously figure plots vca different adoption patterns levels value coordinated information vci defined additional savings achieved coordination top vca presence forecast errors given day coordinator must generate forecasts let independently identically distributed across coordinator forecasting aggregate quantities error lower individual household normally distributed fit scaling law forecasting model normally distributed let coordinator solves linear program except replaces constraint optimal schedule solution coordinator schedule charging discharging collective storage let true net load group cost electricity day let cfwd vci forecast error adoption level vcifwd case size net net storage capacity kwh kwh kwh kwh kwh table five sensitivity study cases evaluated net means household system sized course year generates much energy household inflexible load leads variation size panels among households shape distribution basically line mean load given fig maximum charging discharging rates kwh storage device scaled kwh device parameters maximum charging discharging rates kwh device based second generation tesla powerwall cfwd vcafwd vci reverse random adoption patterns defined analogously figure plots vci different adoption patterns forecast error levels adoption levels bbl sensitivity studies conducted five sensitivity studies subset households listed table qualitative results preserved cases case major difference vca forward adoption monotonically increasing maximum adoption capacities relatively small case storage even last adopter redirected high value applications households offsetting peak period energy consumption cases households larger inflexible loads larger systems thus absolute annual savings distribution uneven largest energy consumers saving much compared households hand normalized savings scaled household consumption even across population point almost households policy relatedly ordering households normalized savings policy weakly correlated orderings policies thus pricing policy significant impact households best normalized savings metric data availability pricing solar irradiance data publicly available smart meter data property obtained authors agreement preserve privacy resampled versions data underlying figures made available editors reviewers data underlying figures made available original form analysis code made available editors reviewers addition reviewers able modify simulation parameters evaluate altered results references california state assembly assembly bill online available https bill akorede hizam pouresmaeil distributed energy resources benefits environment renewable sustainable energy reviews vol online available http jain qin rajagopal planning distributed energy resources amidst complexities nature energy vol kok widergren society devices integrating intelligent distributed resources transactive energy ieee power energy magazine vol may ferreira carvalho ferreira ilic distributed energy resources integration challenges networks voltage control limitations risk cascading ieee transactions sustainable energy vol jan griffith golmie moulema towards integrating distributed energy resources storage devices smart grid ieee internet things journal vol parag sovacool electricity market design prosumer era nature energy vol bayram ustun survey behind meter energy management systems smart grid renewable sustainable energy reviews vol online available http agnew dargusch effect residential solar storage centralized electricity supply systems nature clim change vol kwac flora rajagopal household energy consumption segmentation using hourly data ieee transactions smart grid vol jan darghouth barbose wiser impact rate design net metering bill savings distributed residential customers california energy policy vol online available http borenstein electricity rate structures economics solar could mandatory rates undermine californias solar photovoltaic subsidies bright power energywiz association energy affordability solar realtime pricing pricing beneficial solar new york city new york city economic development corporation tech mills wiser barbose golove impact retail rate structures economics commercial photovoltaic systems california energy policy vol online available http vangeet brown blair mcallister solar san diego impact binomial rate structures real systems yang balducci economic analysis optimal sizing battery storage ieee power energy society general meeting pesgm july peterson whitacre apt economics using hybrid electric vehicle battery packs grid storage journal power sources vol online available http nottrott kleissl washom storage dispatch optimization combined storage systems ieee power energy society general meeting july huang lin economic analysis hybrid photovoltaic battery energy storage system ieee transactions industry applications vol jan stadler aki firestone lai marnay siddiqui distributed energy resources optimization commercial buildings electric thermal storage technologies lawrence berkeley national laboratory tech stadler marnay siddiqui lai aki integrated building energy systems design considering storage technologies lawrence berkeley national laboratory tech neubauer simpson deployment energy storage demand charge reduction national renewable energy lab nrel golden united states tech hoff perez margolis maximizing value systems using storage controls solar energy vol online available http riffonneau bacha barruel ploix optimal power flow management grid connected systems batteries ieee transactions sustainable energy vol july energy information administration electric power monthly tech dec electric power monthly tech dec deutsche bank markets research welcome age may nykvist nilsson rapidly falling costs battery packs electric vehicles nature clim change vol bronski creyts crowdis doig glassmire guccione lilienthal mandel rader seif economics load defection systems compete traditional electric matters possible paths forward rocky mountain institute tech rizy kueck adaptive voltage control distributed energy resources algorithm theoretical analysis simulation field test verification ieee transactions power systems vol aug kim norford analysis experimental implementation grid frequency regulation using batteries compensating fast load demand variations ieee transactions power systems vol jan sevlian rajagopal short term electricity load forecasting varying levels aggregation arxiv preprint april ito consumers respond marginal average price evidence nonlinear electricity pricing american economic review vol borenstein electricity price consumers respond residential demand elasticity pricing preliminary draft april mcallister solar adoption energy consumption residential sector california berkeley university california public utilities commission nov fact sheet residential rate reform online available http public meetings pacific gas electric company electric schedule online available https scheds gtm squared gtm squared annual survey report tech feldman brockway ulrich margolis shared solar current landscape market potential impact federal securities regulation national renewable energy lab nrel golden united states tech solar energy industries association solar photovoltaic technology presentation transcript online available http national oceanic atmospheric administration national climatic data center climate reference network site information handbook tech wilcox national solar radiation database update user manual national renewable energy laboratory nrel golden tech national solar radiation data base individual files online available http nsdrb data viewer online available https
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nonparametric bayesian posterior contraction rates scalar diffusions data kweku abraham feb abstract consider inference scalar diffusion model dxt dwt discrete data periodic coefficients given prove general theorem detailing conditions bayesian posteriors contract around true drift function frequentist minimax rate logarithmic factors besov smoothness classes exhibit natural nonparametric priors satisfy conditions results show bayesian method adapts unknown sampling regime unknown smoothness introduction consider scalar diffusion process starting evolving according stochastic differential equation dxt dwt standard brownian motion considerable interest estimate parameters arbitrary functions place assumptions form model naturally nonparametric explain section problems estimating essentially decoupled setting considered paper consider estimation drift function diffusion coefficient assumed given realistic assume observe full trajectory rather process sampled discrete time intervals estimation problem studied extensively minimax rates attained two sampling frameworks fixed asymptotics taken see reiss asymptotics taken typically assuming also see hoffmann comte see also papers addressing nonparametric estimation diffusions typical frequentist methods one must know sampling regime data drawn particular estimator consistent setting numerical simulations suggest attain minimax rate see discussion chorowski estimators even consistent data previous result known author regarding adaptation sampling regime nonparametric setting found chorowski able estimate diffusion coefficient drift obtains minimax rate derivative smoother volatilities paper consider estimation parameters diffusion model nonparametric bayesian perspective bayesian methods diffusion estimation implemented practice see papaspiliopoulos bayesian estimation statistician need specify prior estimating diffusions discrete samples prior need reference sampling regime bayesian methodology provides natural candidate unified approach settings results imply bayesian methods adapt sampling regime also unknown smoothness drift function see remarks proposition proposition respectively details results proved frequentist assumption fixed true parameter paper belongs field frequentist analysis bayesian procedures see example ghosal van der vaart introduction field previously shown setting posterior contraction rate guaranteeing posteriors corresponding reasonable priors concentrate mass neighbourhoods true parameter shrinking fastest possible rate log factors see nickl complete proof posteriors contract rate adapting sampling regime remains prove corresponding contraction rate setting forms key contribution current paper prove large class reasonable priors exhibit posterior contraction optimal rate log factors turn guarantees point estimators based posterior achieve frequentist minimax optimal rate see remark theorem lowfrequency regimes broad structure proof inspired use testing approach der vaart coupled insight nickl one may prove existence required tests finding estimator good enough concentration around true parameter main ingredients concentration inequality frequentist estimator construct tests true set suitable sufficiently separated alternatives see section small ball result relate distance see section though structure reflects details different estimators lowfrequency setting typically based mixing properties viewed markov chain spectral structure transition matrix see fail take full advantage local information one sees instead use estimator introduced comte uses assumption view estimation regression problem prove estimator concentrates depends key insight paper markov chain concentration results used setting give worse bounds must supplemented type continuity results crucially rely assumption martingale concentration results similarly small ball result setting depends markov chain mixing instead adapt approach van der meulen van zanten demonstrate divergence discrete setting controlled corresponding divergence continuous data model key new result current paper control extended bound variance log likelihood ratio described key attraction bayesian method allows unified approach regimes another attraction naturally suggests uncertainty quantification via posterior credible sets contraction rate theorems proved paper enough prove credible sets behave advertised one may aim nonparametric mises result see example castillo nickl posterior contraction rate proved constitutes key first step towards proof mises result sampled diffusion model since allows one localise posterior around true parameter proofs nickl inverse problem comparable problem framework assumptions notation introduced throughout paper gathered appendix work scalar diffusion process starting evolving according stochastic differential equation dxt dwt standard brownian motion parameters assumed also assume following given continuity guarantees existence upper bound assumption cper assume existence lower bound denotes functions periodic boundary conditions cper assumption continuously differentiable given norm bound precisely assume cper kcper arbitrary given constant note particular upper bounds lipschitz continuous constant maximal set prove contraction general make stronger assumption fact denotes periodic besov space denotes associated norm see secwhere tion definition periodic besov spaces use readers unfamiliar besov spaces may substitute space mildly weaken results generally assume regularity index unknown results therefore aim adaptive least smoothness index fully adaptive would need adapt also assumptions unique strong solution see example bass theorem moreover solution also weakly unique unique law satisfies markov property see proposition theorem denote law cylindrical unique solution started consider data sampled solution asymptotics taken suppress subscript simply write throughout write shorthand data similarly write denote set example constant depending parameters beyond guaranteeing existence uniqueness solution assumptions also guarantee existence invariant distribution periodised process mod defining probability density invariant distribution given eib eib see bhattacharya equations note chosen different normalisation expressions appear slightly different observe bounded uniformly away zero infinity exist constants depending precisely see deduce take assume follows invariant distribution assumption write law full process assumptions write expectation according law note invariant nevertheless function see proof theorem since estimating function implicit assumption unimportant finally need assume fast enough rate precisely use following assumption log unknown constant since already assume new assumption equivalent log constant throughout make frequentist assumption data generated according fixed true parameter denoted use shorthand similarly context allows simply write generic drift remarks comments assumptions assuming given observe continuous data known exactly atr least point visited process via expression quadratic variation hxit data perfectly reconstruct diffusion coefficient data estimate much faster rate drift assumed unknown minimax errors respectively scale shown adapting proofs slightly apply periodic setting use since assume follows large hence estimate faster rate regardless relative smoothnesses note problems estimating setting essentially independent example smoothness affect rate estimating see therefore substantially simplifying problem estimating assumption given assuming known bound kbkcper assumption one derivative typical minimal assumption ensure diffusion equation strong solution solution invariant density assumption given bound cper function undesirable needed proofs particular ensuring uniform lower bound invariant densities lower bound essential markov chain mixing results since reciprocal controls mixing time theorem plausible needing assumption inherent problem rather artefact proof methods bypass markov chain mixing arguments martingale approach lemma also rely uniform lower bound moreover lower bound scales unfavourably boundedness assumptions principle exclude gaussian priors computationally attractive practice one could choose large value approximate gaussian priors arbitrarily well using truncated gaussian priors assuming shown see proof theorem law converges exponential rate starting distribution assuming restrictive mentioned implicit choice rather unit interval arbitrary unimportant since estimate function assuming log typical setting assume indeed minimax rates proved assumption technical reasons concentration section section need slightly strenghtened assumption spaces approximation throughout depend family function spaces purposes take periodised wavelet spaces span denote convenience denote inner product one definition besov norm flk wavelet coefficients sup flk defined cper norm finite see nickl sections construction periodised wavelets proof wavelet norm characterisation agrees possible definitions desired besov space therefore immedinote orthonormality wavelet basis means flk ately follows definition besov norm kkbkb constant projection onto remarks uniform convergence wavelet series wavelet projections converge supremum norm moreover convergence actually uniform follows applying proposition uniformly bounds kbkcper regular boundary regularity functions periodic besov space denoted boundary sense weak derivatives order alternative approximation spaces key properties need approximation spaces first needed using spaces specify priors see section corresponding inequality holds many function spaces replace dim example set trigonometric polynomials degree provided smax given smax generated periodised daubechies wavelets priors built using spaces achieve posterior contraction rate main contraction theorem let prior probability distribution subsets given assume follows law described section write transition densities recall use shorthand assume mapping measurable borel standard arguments example chapter shown bayesian posterior distribution given data introduce shorthand joint probability density data main result paper following theorem designed apply adaptive sieve priors theorem designed use smoothness parameter known see section explicit examples results use theorem consider data sampled solution assumptions let true parameter law full path let sieve prior let periodic wavelet space resolution described section suppose constants projection onto dim constant log probability suppose known let djn positive constants djn let sequence priors satisfying log constant djn achieve rate contraction log probability remark optimality minimax lower bounds strictly apply assumed given nevertheless minimax rate model follows adapting arguments continuous data case kutoyants section apply periodic model observing data outperform continuous data since contraction rate guarantees existence estimator converging true parameter rate example centre smallest posterior ball mass least see theorem ghosal van der vaart rates attained theorem optimal log factors proof given section explicit examples priors results guarantee following priors exhibit posterior contraction proofs found section technical convenience employ slightly stronger assumption simply lying precisely recalling form family wavelets section denotes constant function assume following assumption constant assume sequence specified explicit priors prove contraction random wavelet series priors let iid ulk density satisfying constant constant assumption example one might choose density unif random variable truncated gaussian density define prior law associated random wavelet series ulk assumption give three examples priors built example basic sieve prior let let probability distribution described theorem example normalising constant let proposition preceding prior meets conditions theorem satisfying assumption used define prior appropriate constant thus also constant constant log remark adaptive estimation assume smin assumption automatically holds constant smin thus contrast results prior adapts unknown range smin known fix rate decay wavelet coefficients ensure draw prior lies hand rather relying hyperparameter choose right resolution wavelet space demonstrate following example example known smoothness prior let let define sequence priors take genuine prior sequence priors also work provided fast enough rate proposition assume bounded away zero preceding sequence priors meets conditions theorem satisfying assumption used define prior appropriate constant thus also constant constant log remark assumption mildly stronger assumption wavelet characterisation minimax lower bounds hold slightly smaller class functions satisfying assumption except possible log factor induced weighting factor example prior invariant density applications may natural place prior invariant density implicitly drift function minor adjustments theorem still applied priors outline necessary adjustments identifiable therefore introduce identifiability constraint could fix positive constant reduce case translation choose simplicity assumption standard periodic model example see van waaij van zanten restriction normalising constant log place assumption need similar assumption log precisely assume hlk assumed known put prior proposition since order ensure assume last two ensure assume kcper conditions bypassed careful statement theorem treatment bias proposition make changes listed log remarks adaptation sampling regime essentially shown bayesian posterior adapts unknown sampling regime strictly one must show priors also contract setting since model assumptions differ however one apply standard contraction rates markov chains use results translate contraction invariant density contraction minor technical changes also needed use periodic diffusion use reflected diffusion note given setting minimax rate estimating example see faster rate attained unknown improvement possible one bypasses delicate interweaving problems estimating remains interesting open problem simultaneously estimate method adapts sampling regime adapting proofs paper case unknown would show bayesian method fulfils goal key difficulty making adaptations arises small ball section section girsanov theorem apply diffusions different diffusion coefficients intermediate sampling regime strictly speaking demonstrate robustnestness sampling regime extreme cases fixed author aware papers addressing intermediate regime slower rate minimax rates even known bayesian method attain correct rates intermediate regime log factors remains find explicitly rates attained posterior prove corresponding lower bounds find rates attained posterior one could interpolate markov chain concentration inequalities theorem martingale continuity based inequalities lemmas construction tests section construct tests needed apply general contraction rate theory main result section following recall periodic wavelet space resolution described section projection onto dim lemma let sampled diffusion satisfying grant assumptions let suppose assume dln sequence satisfying log constant exist tests functions data sufficiently large max sup proof given section straightforward consequence constructing estimator appropriate concentration properties first introduce general concentration results need general concentration results use three forms concentration results building blocks theorems first comes viewing data markov chain applying markov chain concentration results results similar used case need track dependence constants second form apply case use quantitative form continuity diffusion processes inequality third form based martingale properties introduced needed lemma markov chain concentration results applied diffusions main concentration result arising markov structure following theorem exists constant depending sufficiently large bounded functions nkf exp min equivalently max space functions indexed subset vector space supf supf also sup max constant proof application following abstract result markov chains theorem paulin proposition theorem let markov chain taking values transition kernel invariant density suppose uniformly ergodic constants transition kernel total variation norm signed measures write tmix min suppose bounded let var let exp proof theorem first note since assumed periodic see mod viewed diffusion circle denote transition densities lemma tells density density satisfying exp sup regularise extend result also applies initial distribution point mass density hence result applies show exp moreover note also note upper bound constant precisely take thus see constant follows fixed discrete time markov chain uniformly ergodic mixing time tmix log constant theorem applies tell exp exp since assumption see constant using bound min upper bounding centred moments uncentred moments deduce result obtained change variables supremum result use standard chaining argument baraud theorem use place baraud assumption noting baraud uses assumption prove expression mirroring rest proof follows exactly precisely following proof take remark proof simplifies restrict satisfying case invariant density upon changing normalising constant reduces familiar form eib diffusion reversible case use theorem instead theorem attain results better constants continuity properties diffusions diffusions realised rescaled brownian motions realising diffusions way allows see inherit form continuity brownian motion note continuity tell changes sufficiently small time steps hence applied regime contrast theorem also applies setting writing log log defining key result section following lemma let solve scalar diffusion equation grant assumptions constant sufficiently large initial value sup constant remark proof lemma sufficiently large bigger max log constant remarks need control increments simultaneously hence include parameter take time horizon applying result simply controlling using union bound give sharp enough results lemma applies distribution point masses application tower law proof strategy build corresponding result brownian motion via time changes rescaling brownian motion well known modulus continuity log exists almost surely log sufficiently close need quantitative version statement supplied following result proved appendix using gaussian process techniques lemma let standard brownian motion universal constant sufficiently large sup time change able deduce following lemma let local martingale quadratic variation satisfying constant constant depending sufficiently large sup particular result applies solution dyt dwt provided remark proof lemma sufficiently large bigger max log constant proof theorem rogers williams write standard brownian motion quadratic variation define event sup wam note sufficiently large occurs probability least universal constant lemma event see sup sup wam wam used wam increasing range attain final inequality easy see recalling assumed wam wam provided holds relevant range thus considered range log log constant depending note absorbed term depending log constant using log desired result follows upon relabelling since allow depend particular case dyt dwt simply observe proof lemma set exp scale function let inverse since exists continuous formula applies yield dyt dwt let max exp observe thus event sup occurs probability least constant lemma lipschitz constant exp follows exp exp result follows upon relabelling exp absorbing constant multiplier concentration drift estimator defining estimator adapt estimator introduced estimator constructed considering drift estimation problem specifically defining dws write note discretization error vanishes takes role noise define empirical norm related empirical loss function kukn leave term notational convenience recalling wavelet space described section define solution minimisation upper bound cper problem argmin chosen choose arbitrarily among minimisers unique main concentration result estimator defined prove following concentration inequality theorem let sampled diffusion satisfying grant assumptions let projection onto sln let defined assume dln dim sln sequence satisfying log uniformly across sufficiently large remark previous proofs bayesian contraction rates using concentration estimators approach see used duality arguments fact supv demonstrate linear estimators considered satisfy concentration inequality desired form key insight paper model consider achieve required concentration using minimum contrast estimator see massart need techniques differ substantially duality arguments proceeding proof demonstrate used prove existence tests suitably separated alternatives typical uniqueness since minimiser proof lemma let estimator outlined let let constant uniformly across define test otherwise see immediately sup sup sup proof theorem enough show uniformly across initially considering finding corresponding eliminate factor front exponential indeed enough prove constant multiplier front exponential multiplier convenient structure proof follows uniform upper bound bounds mean equivalent show equivalent empirical norm event sufficiently high probability finally definition estimator allow control empirical distance bkn end write defining introduce following set events sln ktkn constant chosen decompose thus proved theorem completed following show theorem holds deterministically acn large enough constant show suitable choice show choose step intuitively reason follows event acn occur small compared since assumed uniform supremum bound functions fact holds unless small absolute terms small formalise reasoning define chosen note definition since large enough uniformly across deduce since also construction deduce provided thus acn contained event find using using acn deterministically large enough depending step show sufficiently large sufficiently small sup form required apply theorem since indexing set lies vector space dimension dln apply theorem see sup max dln dln constant provided choose max result proved choice made assumed dln step since projection recall note since also deduce dropped indicator functions terms right except need later thus using union bound constant precisely take remains show probabilities right exponentially small bounding bkn show constant bkn sufficiently large since deterministic function enough show different step apply theorem working single function large enough bounds derived applying theorem gives max constant see upper bounded constant multiple hence choosing large enough holds bounding show constant recall application showed sufficiently large hence see lies definition use show bkn bkn recall argument copied sections follows using see expanding rearranging yields repeatedly apply standard inequality yield far therefore shown large enough next recall since deduce right hand side included indicator functions help future steps thus union bound constant take already shown large enough constant thus following lemmas conclude proof lemma conditions theorem constant sufficiently large lemma conditions theorem constant sufficiently large proof lemma recall recall lipschitz continuous lipschitz constant therefore enough bound sup apply continuity result lemma constant lemma noting assumption log ensures large enough compared conditions lemma met least large see sup log log event probability least able take supremum instead large enough note used increases least large enough derived fact log increases range observe log log large enough eventually assumption log able deduce log follows suitably chosen constant independent implies desired concentration proof lemma recall lemma tells dws exp drift function exp apply theorem see constant exp min exp min obtain last line used write min returning find exp min exp min ktk ktk exp min constant show suitably chosen minimum inside exponential reduces min dependent part minimum optimised min working expression one finds natural cases split equivalently dependent part bound optimised overall minimum min dependent part bound optimised assumption ing overall minimum taking form min guarantees first term smaller two cases therefore bound exp min changing variables attain bound max exp theorem standard chaining argument allows deduce sup dln constant taking recalling assumption dln obtain desired result conditional case event small ball probabilities show divergence laws corresponding different parameters controlled terms parameters denote divergence probability distributions densities log log also write log note implicitly depends via recalling density introduce following type neighbourhoods define bkl log log log main result section following theorem let sampled diffusion satisfying grant assumptions assume sequence constant sufficiently large bkl proof applying lemma appendix shown var log var log var log noting also linearity observe bkl therefore enough show follows immediately applying lemma substitution lemma conditions theorem sufficiently large key idea proving lemma use divergence laws paths control divergence help calculate divergence full paths using girsanov theorem gives explicit formula likelihood ratios let denote law conditional restriction write throughout section simply write similarly following theorem girsanov theorem assume satisfies assumption laws mutually absolutely continuous continuous path dpb exp dut proof see theorem noting assumptions met lipschitz bounded write dpb derivatives densities respect whose existence girsanov theorem guarantees simply write context allows similarly since path omit superscripts densities general writing similarly proof lemma break proof series lemmas upper bound variances definition corresponding second moments step show constant shows whenever content lemma step show log content lemma note steps need assumptions step uses step show constant max log follows log content lemma together three lemmas conclude proof lemma conditions theorem constant proof follow proof given lemma adapting expanding steps let let probability density transition respect lebesgue density transitions calculated integrating likelihood paths start end performing integration exactly yield conditional expectation given say similar expression holds log lemma appendix allows simplify ratio conditional expectations says mutually absolutely continuous probability distributions measurable function apply deduce log log used conditional jensen inequality deduce final line side divergence laws continuous paths different drift parameters averaged starting distribution introducing notation evaluate applying girsanov theorem fubini theorem log dxt dwt final line dropped one terms using dws martingale bounded semimartingale integrated square integrable martingale yields martingale note bounded bounded bounded away zero since periodic replace final expectation periodised version mod expectation respect law started invariant distribution thus deduce replace denominator lower bound note assumptions ensure equivalent constants depending result follows lemma conditions theorem constant sufficiently large log whenever proof wish apply similar argument previous proof hindered fact log convex unlike log directly able apply jensen inequality therefore instead consider smallest convex function dominating log given log fact convenient equivalent think dominating function log find control second moment log second moment corresponding expression log full paths approximation error small small log apply lemma jensen inequality tower law find log conditional thus promised decomposition corresponding quantity continuous case approximation error log log conclude showing two terms bounded provided sufficiently small constant showing log write apply girsanov theorem theorem find log dwt dwt dwt recall dwt martingale since bounded note finite variation process follows third term right final expression vanishes see first term right use isometry fubini theorem periodicity stationarity periodised process mod find dwt upper bounded bound since term dominated large second term find log seen uniformly bounded thus constant usual assumptions allow upper bound constant log showing use standard trick expressing expectation terms tail probabilities lemma shown log log constant tower law also holds hence following log log log noting write integrating parts easy show log log log log log log log least large enough denominator positive follows sufficiently large log dominated term far shown log recall aim show expectation bounded hence done prove large enough log since greater sufficiently large enough show log recall assume follows large enough log log log observe log log log log log log required lemma conditions theorem constant max log proof comment lemma suffices prove ckb hellinger distance densities defined since uniformly bounded away zero absorb term constant initially prove pointwise bounds difference densities recall saw section eib eib decompose eib eib bounds eib application mean value theorem tells eib exp exp constants expression upper bounds follows constant ckb bound holds since hbb similar decomposition yields thus shown integrating pointwise bound find finally since different constant done main contraction results proofs tools need apply general theory order derive contraction rates first deduce following abstract contraction result demonstrate theorem follows recall denotes divergence probability distributions densities recall definition bkl log theorem let sampled diffusion satisfying grant assumptions let sequence positive numbers let sequence positive integers constant dln log let assume projection onto sln described section let sequence priors satisfying bkl constant probability fact replace exponent choose simplifies exposition exact value unimportant main step proof demonstrate evidence lower bound standard proof relegated appendix lemma evidence lower bound elbo recall defined bkl log joint probability density started invariant distribution true parameter denotes let write bkl bkl define event acn bkl proof theorem write enough show observe measurable sets event function decompose apply given statement theorem test given lemma noting assumptions theorem include needed lemma take expectation bound terms separately bounding bounding acn lemma since assumption bkl expectation tends zero lemma bounding since apply fubini theorem see since assumed deduce exp bounding similar argument observe construction tests see integrand bounded choosing large enough could attain fixed exponential term choosing corresponding see proof theorem apply theorem key idea allows control bias obtain adaptive result sieve prior undersmoothing specifically prove small ball probabilities conditioning hyperprior choosing resolution corresponds minimax rate rather corresponding slower rate log prove contraction logarithmic gap gives room need ensure achieve bias condition small ball condition constant argument goes follows write let log choose natural numbers satisfying least large enough djn dln constant chosen note holds definition recall choice approximation spaces section fixed therefore find large enough writing log log similarly shown constant small ball result theorem large enough set sln observe calculation shows bias condition holds since also see remains therefore show assumptions theorem deduce constant bkl recall theorem tells bkl large enough thus suffices show also achieve observe sjn next note thus see using assumptions sjn djn exp djn djn log log exp log constant using log log log log log see log different constant changing constant absorb factor exponential large enough since assumption slcn assumed ensures sum constant times dln proved therefore choose large enough guarantee let statement theorem define take similarly apply results section see sufficiently large set assumptions guarantee bias condition hold indeed thus suffices prove constant since absorb factor constant substituting prior concentrates see thus argument similar previous part indeed slightly simpler omit remaining details explicit priors proofs proof proposition verify prior satisfies conditions theorem condition holds construction wavelet characterisation since max see section see assumption choice ensures follows standard besov spaces results proposition adapted apply periodic besov spaces cper per bounded terms thus appropriate choice similarly see remains show holds ulk max max since assumed follows independence ulk side last expression lower bounded holds place proof proposition verify conditions theorem throughout proof simply write since appropriate choice proof proposition observe also construction using wavelet characterisation thus holds remains check let similarly proof proposition djn required proof proposition include key differences previous proofs choice seen norm bounded appropriate constant wavelet characterisation flk seen lie hence appropriate constant also note hkb similarly observe previous proof hkb hkb log large enough similarly thus final inequality seen hold wavelet representation using small ball condition follows updated also assumptions appendix remaining proofs technical lemmas proof lemma follow adapt proof dudley theorem theorem gaussian process indexed intrinsic covariance metric dudley theorem says sup log number balls needed cover defining process log shown proof gaussian process bounded continuous sample paths follows theorem sup upper bound applying trivial lower bound denominator log log follows fact less half diameter using also intrinsic covariance metric deduce log take log apply result standard brownian motion intrinsic covariance metric applying jensen inequality see log log log log restrict supremum satisfying observe log log log noting recalling defined log log see log thus writing sups log see log sup since fixed finite number write log sup thus provided larger result constant log remark need know grows order know large enough lemma apply observe write maxk sup log since see sup log whole process apply theorem see log uncentred subgaussian parameter log see lemma constant depending expectation bound maximum standard results subgaussian variables see lemma max log log number take maximum scales linearly follows order bounded log proof lemma write bkl renormalised restriction bkl jensen inequality write bkl bkl exp log bkl log bkl log applying fubini orem using definition bkl see sup log applying jensen inequality twice applying fubini theorem see log bkl log bkl bkl bkl log log log log obtain last line used bound variance log bkl together bounds mean variance tell exp exp applied chebyshev inequality obtain final inequality rightmost expression tends zero since assumption result follows remark true acn tending zero different rate define instead bkl say exact value exponent important proof lemma let mutually absolutely continuous probability measures write measurable function proof follows straightforwardly using characterisation conditional expectation terms expectations functions precisely recall holds function conversely holds version conditional expectation converse statement fact enough hold indicator functions applying repeatedly find since also version required lemma grant assumptions let exists constant initial value log log sufficiently large proof applying girsanov theorem theorem log dwt dwt write dws martingale whose quadratic variation satisfies max since uniformly upper bounds thus apply lemma find provided large enough log sup exp constant depending recall log result follows noting condition large enough lemma replaced large enough lemma variance log likelihood ratio tensorises model constant precisely log log log proof write log log log odd log even note sums independent terms since markov chain thus odd log corresponding result using var var var var cov cov cov cov var var one derives elementary inequality var var var var result follows appendix notation collect notation used course paper solution dxt dwt periodised diffusion mod drift function diffusion coefficient invariant law section initial condition varb according law started notation transition densities respect lebesgue measure density respect true parameter generating data etc shorthand etc lower upper bound constant log maximal paramater space cper cper periodic besov space wavelet approximation space resolution generated periodised meyer wavelets span used notation constant function dim orthogonal projection onto log log indicator set event divergence densities log log bkl log log log prior distribution posterior distribution given data inner product supremum kkcper cper per empirical references baraud inequality suprema random processes applications model selection regression bernoulli doi url http bass stochastic processes cambridge university press bhattacharya denker goswami speed convergence equilibrium normality diffusions multiple periodic scales stochastic processes applications doi url http functions use continuous hence take supremum rather needing essential supremum massart minimum contrast estimators sieves exponential bounds rates convergence bernoulli url https castillo nickl nonparametric mises theorems gaussian white noise annals statistics doi url https castillo nickl mises phenomenon nonparametric bayes procedures annals statistics doi url https chorowski statistics diffusion processes low observations phd thesis berlin doi http comte rozenholc penalized nonparametric mean square estimation coefficients diffusion processes bernoulli doi url http dalalyan sharp adaptive estimation drift function ergodic diffusions ann statist doi url https ghosal ghosh van der vaart convergence rates posterior distributions annals statistics doi url http ghosal van der vaart fundamentals nonparametric bayesian inference cambridge university press ghosal van der vaart convergence rates posterior distributions noniid observations annals statistics doi url https nickl mathematical foundations statistical models cambridge university press doi url http nickl rates contraction posterior distributions annals statistics doi url http gobet hoffmann nonparametric estimation scalar diffusions based low frequency data annals statistics doi url http gugushvili spreij nonparametric bayesian drift estimation multidimensional stochastic differential equations lithuanian mathematical journal doi url https hoffmann adaptive estimation diffusion processes stochastic processes applications doi url http kutoyants statistical inference ergodic diffusion processes springer london doi liptser shiryaev statistics random processes general theory springer berlin heidelberg van der meulen schauer van waaij adaptive nonparametric drift estimation diffusion processes using expansions statistical inference stochastic processes doi url https van der meulen van zanten consistent nonparametric bayesian inference discretely observed scalar diffusions bernoulli doi nickl bernstein von mises theorems statistical inverse problems equation url https nickl nonparametric bayesian posterior contraction rates discretely observed scalar diffusions annals statistics doi url https papaspiliopoulos nonparametric estimation diffusions differential equations approach biometrika doi url http paulin concentration inequalities markov chains marton couplings spectral methods electronic journal probability doi url https pokern stuart van zanten posterior consistency via precision operators bayesian nonparametric drift estimation sdes stochastic processes applications doi https url http ray bayesian inverse problems priors electronic journal statistics doi rogers williams diffusions markov processes martingales ito calculus cambridge university press trabs adaptive confidence bands markov chains diffusions estimating invariant measure drift esaim probability statistics doi url https van waaij van zanten gaussian process methods diffusions optimal rates adaptation electronic journal statistics doi url https
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policy learning miao christopher xuejun lawrence jonathan nov laboratory information decision systems mit cambridge usa miaoliu department computer science university new hampshire durham usa camato department electrical computer engineering duke university durham usa xjliao lcarin department aeronautics astronautics mit cambridge usa jhow abstract expectation maximization recently shown efficient algorithm learning controllers fscs large decentralized pomdps however current methods use fscs often converge maxima far optimal paper considers fsc represent local policy agent fscs constructed using prior leading new framework called decentralized policy representation approach learns controller parameters variational bayesian algorithm without assume model available performance demonstrated several benchmark problems showing algorithm scales large problems outperforming methods introduction decentralized partially observable markov decision processes provide general framework solving cooperative multiagent sequential problems arise numerous applications including robotic soccer transportation extraplanetary exploration traffic control viewed pomdp controlled multiple distributed agents agents make decisions based local streams information observations joint actions control global state dynamics expected reward team decentralized individual agent generally enough information compute global belief state sufficient statistic decision making pomdps makes generating optimal solution difficult pomdp especially long planning horizons circumvent difficulty solving optimally still generating high quality policy paper presents scalable learning methods using finite memory policy representation problems continue infinite number steps significant progress made agent policies represented controllers fscs map observation histories actions recent work shown scalable method generating controllers large addition also shown efficient algorithm reinforcement learning agents learn fscs based trajectories without knowing learning model important yet unanswered question define appropriate number nodes fsc previous work assumes fixed fsc size agent number nodes affects quality policies convergence rate number nodes small fsc unable represent optimal policy therefore quickly converge result contrast number large fsc overfits data often yielding slow convergence policy paper uses bayesian nonparametric approach determine appropriate controller size fsc following previous methods learning assumed centralized execution decentralized learning accomplished offline based available information optimization decentralized solutions controller constructed using prior prior allows number nodes variable set nodes actively used controller encouraged compact nodes actually used determined posterior combining prior information trajectory data framework called decentralized policy representation recognize role prior addition use fscs paper also makes several contributions specifically algorithm directly operates shifted empirical value function decpomdps simpler likelihood functions mixture dynamic bayes nets dbns existing frameworks moreover derive variational bayesian algorithm learning based agents trajectories episodes actions observations rewards algorithm linear number agents square problem size therefore scalable large application domains practice trajectories generated simulator set experiences provided batch data scenario general realistic widely adopted learning demonstration reinforcement learning best knowledge first application bayesian nonparametric methods difficult problem proposed method able generate solutions large problems background related work introducing proposed method first describe model related work decentralized pomdps represented tuple finite set agent indices respectively sets joint actions observations available agent step joint action selected joint observation received set finite world states state transition function denoting probability transitioning taking joint action observation function probability observing taking joint action arriving state reward function immediate reward received taking joint action discount factor global reward signal generated team agents joint actions taken agent observes local observation agent lacks access agents observations agent maintains local policy defined mapping local observation histories actions joint policy consists local policies agents initial belief state objective find joint policy value starting maximized fsc compact way represent policy mapping histories actions formally stochastic fsc agent defined tuple han defined finite set controller nodes agent initial node distribution probability agent initially set markov transition matrices denoting probability controller node transiting agent takes action sees observation set stochastic policies probability agent taking action simplicity use following notational conventions def cardinality follow similarly joint fsc agents variable abbreviated variable index range shown subscript superscript index range obvious context simple used instead thus represents actions agent step represents node transition probabilities agent starting node taking action seeing observation given local history actions observations step well agent controller calculate local policy probability agent chooses action planning inference planning problem transformed inference problem efficiently solved algorithms validity method based fact introducing binary rewards choosing geometric time prior maximizing likelihood mixture dynamic bayes nets equivalent optimizing associated policy value related affine transform rmin rmax rmin rmax rmin def rmax rmin maximum minimum reward values trmin shifted value previous methods achieved success scaling larger problems factoring distribution states histories inference methods require using decpomdp model construct bayes net policy evaluation exact model parameters unknown one needs solve reinforcement learning problem address important yet less addressed problem global empirical value function extended case constructed based action observation reward trajectories product local policies agents serves basis learning fscs settings definition global empirical value function let rtkk set episodes resulting agents choose actions according set stochastic behavior policies action history global empirical def akn min value function defined discount according strong law large numbers large number trajectories empirical value function approximates rately hence applying approximates offers objective learning decentralized policies directly maximized algorithms bayesian learning policies algorithms infer policies based representation observed data difficult explicitly handle model uncertainty encode prior expert knowledge address issues bayesian learning method section accomplished measuring likelihood using combined prior bayes rule yield posterior marginal likelihood joint fsc additive constant proportional marginal value function def compute posterior markov chain monte carlo mcmc simulation straight forward method however mcmc costly terms computation storage lacks strong convergence guarantee alternative variational bayes method performs approximate posterior inference minimizing divergence true approximate posterior distributions method local convergence guarantee able scalability accuracy focus derivation method denoting variational approximation qtk approximation objective function qtk qtk qtk def qtk lower bound def rtk reward since equation independent qtk minimizing divergence equivalent maximizing lower bound leading following constrained optimization problem qtk subject qtk qtk constraint second line arises approximation decentralized policy representation last two lines summarize normalization constraints worth emphasizing developed variational approximation optimize decentralized policy representation showing learning problem formulation general accurate method multiagent problem considered paper policy priors solve bayesian learning problem described obtain fscs prior used specify policy structure formally given definition refer appendix derivation details definition decentralized stick breaking policy representation tuple definition unbounded set nodes indexed positive integers notational assumed deterministic determine fsc parameters defined section follows dir dir represents distribution represents process dirichlet beta gamma decsbpr differs previous nonparametric bayesian methods specifically performs generalizes nonparametric bayesian policy representation pomdps decentralized domain whereas method assume knowledge world model explicitly learns performs planning moreover distinguishes previous methods prior distributions inference methods employed previous methods employed hierarchical dirichlet processes hidden markov models infer number controller nodes however due lack conjugacy two levels dps fully conjugate bayesian variational inference therefore methods used mcmc requires high computational storage costs making ideal solving large problems contrast employs single layer priors fsc transition matrices sparse gamma priors weight hyperparameters bias transition among nodes smaller indices similar framework explored infer hmms refer readers details worth noting processes subsume dirichlet processes dps special case purpose using priors encourage small number fsc nodes compared priors represent richer patterns sparse transition nodes fsc allows arbitrary correlation weights weights always negatively correlated variational policy inference shown random weights constructed prior equivalently governed generalized dirichlet distribution gdd therefore conjugate multinomial distribution hence efficient variational bayesian algorithm learning decentralized policies derived accommodate unbounded number nodes apply retrospective representation priors agent prior set truncation level taking account current occupancy well additional nodes reserved future new occupancies solution priors given theorem proof provided appendix nonparametric priors also used method imposes proposals top level dps lacking uncertainty measure algorithm batch inference input episodes number agents initial policies lower bound iter update global rewards using compute end iter iter compute lbiter using lbiter compute using update using compute using end end return policies controller sizes theorem let constructed priors defined iterative application following updates leads monotonic increase convergence maxima updates qtk qtk set computed using replaced ehln probability mass functions ehln denotes expectations respect distributions posterior distribution updated ptk indicator function marginals update equations theorem constitute algorithm learning joint fscs priors batch data particular step rewards reweighted reflect improved marginal value new policy posterior updated previous iteration step reweighted rewards used improve policy posterior steps require computed based messages updating equations derived appendix determine number controller nodes occupancy node computed checking positive reward assigned example action node reward assigned quantity greater zero node visited summing actions gives value node hence computed based following formula complete algorithm described algorithm upon convergence algorithm point estimates decentralized policies may obtained calculating expectation see appendix details table computational complexity algorithm var best case worst case computational complexity time complexity algorithm iteration summarized table assuming length episode order magnitude number nodes per controller order magnitude table worst case refers nonzero reward every time step episode dense rewards best case nonzero reward received terminal step hence general algorithm scales linearly number episodes number agents time dependency linear quadratic case computational complexity algorithm independent number states making scalable large problems exploration exploitation tradeoff algorithm assumes batch learning trajectories collected using separate behavior policy appropriate data generated simulated experiences without input learning algorithm learning demonstration learning efficient behavior policy close optimal case expert information available guide agents random behavior policy may take long time policy converge optimality case agents may want exploit policies learned far speed learning process important issue concerns keeping proper balance exploration exploitation prevent premature convergence suboptimal policy allow algorithm learn quickly since execution policies decentralized difficult design efficient exploration strategy guarantees optimality count visiting frequency fsc nodes apply style heuristic select next controller nodes use strategy select actions however might sample inefficient proposed distributed learning approach agents take turns learn best response policies framework applies type heuristic using counts trajectories distinguish known unknown histories tradeoff exploration exploitation however method confined policies problems requires synchronized learning better accommodate bayesian policy learning framework define auxiliary fsc represent policy agent balancing exploration exploitation avoid confusion refer primary fsc two components distinguishing encodes exploration exploitation denoting probability agent choosing one express way one expresses described section behavior policy agent given primary fsc policy exploration policy agent usually uniform distribution behavior policy achieved significant success case extend case centralized learning decentralized provide empirical evaluation next section experiments performance proposed algorithms evaluated five benchmark problems problem traffic control experimental procedure used results reported hyperparameters set promote sparse usage fsc policies initialized fscs converted episodes highest rewards using method similar learning fsc learning fsc demonstrate advantage learning fscs compared implementation previous algorithm comparison mars rover problem using episodes learn fscs evaluating policy discounted accumulated reward averaged test episodes steps consider learning apply policy values chosen testing approach robust values using smaller training sample size method still perform robustly shown appendix mars rover mars rover cpu time sec policy value max mean number nodes number nodes figure comparison controller learned controllers learned left testing value right averaged computation time although less computational complexity algorithm per iteration empirically algorithm uses less time reach convergence moreover average value using prior special dirichlet instance close best result dotted line much better average results solid back line using prior achieves slightly better performance explained flexibility prior explained last paragraph section table performance benchmark problems compared algorithms shows policy values higher value indicates better performance cpu times algorithms average controller size inferred policy learning unknown model roblems iger roadcast ecycling robots ushing ars rovers fixed iteration value ime fixed time value ime planning known model mcem value ime eri value ime value ime collect samples specifically learning agent allowed access episodes collected taking actions according pomdp algorithm value iteration pbvi let probability agents follow pbvi policy probability agents take random actions procedure mimics approach used previous work results reported figure shows exact value computation time function number controller nodes expected algorithm small fscs represent optimal policy number nodes large fscs overfits limited amount data perform poorly even set number inferred still suffer severely initialization local maxima issues seen large setting high truncation level employs algorithm integrate uncertainty policy representation prior result infer number nodes needed optimal controller parameters simultaneously furthermore inference done less computation time higher value improved robustness low variance test value value mcem heuristic heuristic mcem heuri heuri number iterations number iterations states agents figure performance comparison traffic control problem inferred controller size right function algorithmic iteration test reward left comparison methods performance also compared several methods including monte carlo mcem similar mcem approach apply strategy described section follow experimental procedure report rewards running fixed number iterations fixed amount time summarized respectively table first column category shown achieve better policy values mcem problems results explained fact sensitive initialization prone local optima moreover fixing size controllers optimal policy algorithms might fitted using bayesian nonparametric prior learns policy controllers allowing flexibility representing optimal policy also show result running amount clock time mcem fixed time indicates achieve better policy value learning time mcem finally compared periodic periem two planing methods known models generating controllers model allows accurate value function calculations finite number trajectories value periem treated methods approach sometimes outperform periem produces lower value method showing produce near optimal solutions problems produces solutions much closer optimal previous methods also worth noting neither periem scale large problems one discussed using approach scale well scaling larger domains demonstrate scalability large problem sizes large numbers agents test algorithm traffic problem states agents controlling traffic flow intersections one agent located intersection except mcem previous algorithms able solve large learning curves shown appendix results provided personal communication authors run benchmarks available online problems since authors use policy comparing traffic flow two directions heuristic generating training trajectories also use heuristic fair comparison addition examine effectiveness strategy described section also consider case initial behavior policy random optimized discussed figure see help heuristic achieve best performance without using heuristic using strategy iterations able produce higher quality policy mcem moreover inferred number fsc nodes averaged agents smaller number preselected mcem shows scale large problems also produce solutions methods large problems conclusions paper presented scalable bayesian nonparametric policy representation associated learning framework generating decentralized policies new method extends popular method also provided reinforcement learning experimental results show produces solutions method additional benefit inferring number nodes needed represent optimal policy resulting method also scalable large domains terms number agents problem size allowing policies large learned efficiently data acknowledgments acknowledgments research supported office naval research onr muri program award nsf award appendices proof theorem variational bayesian inference standard variational theory minimizing divergence equivalent minimizing lower bound log marginal likelihood empirical value function case using jensen inequality obtain following lower bound log marginal value function qtk qtk qtk qtk def rtk assume accommodate decentralized policy representations derive updating equations rewrite lower bound equation follows qtk qtk ztk qtk pkt inference algorithm based maximizing distribution joint parameters achieved alternating following steps update distribution nodes keeping fixed solve max qtk subject normalization constraint qtk step construct lagrangian fqtk take derivative qtk set result zero qtk qtk ztk solved give distribution nodes nth agent exp akn hln exp hln hln exp exp hln exp hln exp hln exp hln hln ean exp exp hln exp hln hln hln digamma function reward allocated transition state posterior parameters update sufficient statistics step step distribution nodes qtk fixed objective solve maxq subject normalization constraint first consider finding end construct lagrangian exp akn exp akn akn okn relation weights independent beta random variables dir gdd ptk marginalized reward computed equation find construct lagrange qtk exp exp exp hln yyy yyy vjn exp vjn one set case approximation product independent gamma distributions however longer gamma distribution prior likelihood conjugate solve issue one might consider inference method priors consider point estimate maximized one way obtain maximum estimate solve however operation involves taking derivative gamma functions simple form solution circumvent difficult use grid search make search efficient use bounds give initial estimate searching range bounds wendel double inequality basics priors provide definition prior sbp connection generalized dirichlet distribution corresponding posterior inference well main statistics characteristics useful developing inference methods paper detailed mathematical treatment readers referred definition priors almost surely discrete random probability measures measurable space partitioned disjoint regions expressed weights independent beta random variables sbp allows rvs atoms associated resulting weights drawn simultaneously prior generalized dirichlet distribution denote constructing infinite process gdd finite gdd stands generalized dirichlet distribution see connection sbp gdd set truncation level number occupied states write density function changing variables using relation described density obtained follows mean variance element keeping gdd equivalent standard dirichlet distribution concrete example consider case plugging relations obtain bayesian inference gdd given set discrete observations discrete prior posterior written pvi vivi gdd posterior updated indicator function value equal one argument true zero otherwise computation forward backward variables forward backward variables defined section similar messages algorithm hidden markov models variables computed recursively agent using additional experimental results learning fsc learning fsc additional experiments added study impact number training samples mars rover mars rover max mean policy value policy value mars rover max mean number nodes mars rover number nodes number nodes mars rover cpu time sec cpu time sec max mean policy value number number nodes figure comparison controller learned byof nodes controller learned algorithm top testing value policies learned using different number training episodes left middle right bottom averaged computation time left right sequential batch learning exploration exploitation additional results sequential batch learning exploration exploitation five domains plotted figure iteration batch samples collected updated behavior used learn set new policies algorithm results associated numbers reported first two columns table main body paper generate plots trajectories collected iteration exploration parameter set figure additional plots illustrating tradeoff including testing value left inferred controller numbers middle exploration rate right iteration batch samples collected updated behavior policies used learn set new policies algorithm references benchmarks http amato bernstein zilberstein optimizing stochastic controllers pomdps decentralized pomdps autonomous agents systems amato chowdhary geramifard ure kochenderfer decentralized control partially observable markov decision processes proc conf decision control amato konidaris cruz maynor kaelbling planning decentralized control multiple robots uncertainty icra amato zilberstein achieving goals decentralized pomdps proc int conf autonomous agents multiagent systems banerjee lyle kraemer yellamraju sample bounded distributed reinforcement learning decentralized pomdps proc aaai conf artificial intelligence beal variational algorithms approximate bayesian inference phd thesis university london bernstein zilberstein washington bresina planetary rover control markov decision process int symposium artificial intelligence robotics automation space bernstein amato hansen zilberstein policy iteration decentralized control markov decision processes artificial intelligence research bernstein givan immerman zilberstein complexity decentralized control markov decision processes mathematics operations research bishop pattern recognition machine learning volume springer new york bryant sudderth truly nonparametric online variational inference hierarchical dirichlet processes advances neural information processing systems cai liao carin learning explore exploit pomdps proc annual conf neural information processing systems dempster laird rubin maximum likelihood incomplete data via algorithm royal statistical society dibangoye buffet charpillet approximations discounted decentralized pomdps machine learning knowledge discovery databases springer wingate roy tenenbaum nonparametric bayesian policy priors reinforcement learning advances neural information processing systems ferguson bayesian analysis nonparametric problems annals statistics ishwaran james gibbs sampling methods priors statist kumar zilberstein anytime planning decentralized pomdps using expectation proc conf uncertainty artificial intelligence liao carin reinforcement learning partially observable stochastic environment machine learning research liu liao infinite regionalized policy representation proc int conf machine learning liu liao carin online expectation maximization reinforcement learning pomdps proc int joint conf artificial intelligence martins demiris learning multirobot joint action plans simultaneous task execution demonstrations proc int conf autonomous agents multiagent systems messias spaan lima planning uncertainty communication case study fifth workshop sequential decision making uncertain domains msdm oliehoek decentralized pomdps reinforcement learning state art adaptation learning optimization pages springer berlin heidelberg berlin germany paisley carin hidden markov models priors signal processing ieee trans pajarinen peltonen periodic finite state controllers efficient pomdp planning proc annual conf neural information processing systems papaspiliopoulos roberts retrospective markov chain monte carlo methods dirichlet process hierarchical models biometrika pineau gordon thrun value iteration anytime algorithm pomdps proc int joint conf artificial intelligence losonczi bounds ratio two gamma functions journal inequalities applications rabiner tutorial hidden markov models selected applications speech recognition proceedings ieee robert casella monte carlo statistical methods volume citeseer wang blei variational inferecne nonconjugate models journal machine learning rsearch wong generalized dirichlet distribution bayesian analysis applied mathematics computation zilberstein jennings expectation maximization decentralized pomdps proc int joint conf artificial intelligence
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aug partitioning large scale deep belief networks using dropout sai zhang university washington szhang yanping huang university washington huangyp abstract deep learning methods shown great promise many practical applications ranging speech recognition visual object recognition text processing however current deep learning methods suffer scalability problems applications forcing researchers users focus smallscale problems fewer parameters paper consider machine learning model deep belief networks dbns yielded impressive classification performance large number benchmark machine learning tasks scale dbn propose approach use computing clusters distributed environment train large models dense matrix computations within single machine sped using graphics processors gpu training dbn machine randomly drops portion neurons hidden layer training case making remaining neurons learn detect features generally helpful producing correct answer within approach developed four methods combine outcomes machine form unified model preliminary experiment mnist handwritten digit database demonstrates approach outperforms state art test error rate introduction deep learning methods aim learn multilayer neural network extract feature hierarchies input data maximizing likelihood training data promise largely lies potential use vast amounts unlabeled data learn complex highly nonlinear models millions parameters recent competitions deep learning methods shown adavantanges based shallow methods kernel methods ensemble methods paper consider machine learning model deep belief networks dbns learn hierarchical representations inputs dbn applied number machine learning applications including speech recognition visual object recognition text processing among others particular dbn especially problems inputs infer rich models many hidden layers example applied images dbn easily tens millions free parameters ideally would want use millions unlabeled training examples richly cover input space demonstrated increasing scale deep learning respect number training examples number model parameters drastically improve ultimate classification accuracy unfortunately current algorithms even training dbn take weeks using conventional implementation single cpu primarily due daunting computational requirements dbn training large number parameters need trained available examples address dbn scalability problem paper proposes approach scale deep belief networks dbns adapting idea random dropout random dropout proposed hinton originally used prevent complex training data single processor training case hidden unit randomly omitted network probability hidden unit rely hidden units present many separate dbns trained applied independently test data reduce predication bias single dbn approach extends random dropout idea distributed parallel setting rather omitting hidden unit probability approach randomly drops portion hidden units processor training case combine dbns processor approach offers four different ways section performing model averaging trained dbns using majority vote predication result trained dbn test case processor synchronously updating parameters fetching needed parameters processors processor asynchronously fetching computed parameters processors pushing computed parameters processors validated preliminary evaluation using random dropout approach outperforms dbn algorithms data set potential exhibit nearly linear speedup parallel implementation paper makes following contributions approach propose approach scale deep belief networks using random dropout large clusters section implementation implemented approach prototype publicly available http evaluation applied approach mnist dataset demonstrated effectiveness section related work recently many approaches developed scale machine learning algorithms within machine via multithreading across machines via message passing much existing work focuses linear convex models takes distributed gradient computation first step approaches relax synchronization requirements exploring delayed gradient updates convex problems exploring asynchronous stochastic gradient descent architectures single machines another way scale machine learning algorithms provide better abstractions wellencapsulated computation tools mapreduce graphlab two notable examples however mapreduce originally designed parallel data processing number limitations training deep belief network hand graphlab designed general unstructured graph computations exploit computational effectiveness typical structured graph deep belief network thus still unknown whether abstraction graphlab used training dbns deep learning community work done train relatively small models single machine general training model computationally intensive thus full model parallelism well smart distributed optimization techniques required recent years saw surge interest scaling training inference algorithms used dbns improving applicable optimization procedures existing approaches primarily fall following two categories approaches first category use graphics processors gpus achieve significant speedup training dbns use gpus significantly reduced computation time matrix operations dominate computation cost deep learning algorithms however known limitation approaches speedup small model fit gpu memory typically less gigabytes thus effectively leverage gpu researchers often reduce model size parameter number alleviate impact lacking enough gpu memory data parameter reduction work well small problems acoustic modeling speech recognition less attractive realistic problems large number examples dimensions images approaches second category use model parallelism achieve scalability example distbelief notable framework enables model parallelism across machines via message passing details parallelism synchronization communication managed framework model parallelism distbelief framework suffers large communication overhead due dense connections layers neurons data parallelism also supported distbelief using multiple replicas model optimize single objective however pointed hinton large neural network one trained distbelief still perform poorly test data relationship input correct output complicated network enough hidden units model accurately cases typically many different settings weights model training set almost perfectly weight vectors make different predictions test data almost worse test data training data feature detectors tuned work well together training data test data approach inspired mentioned approaches aims address limitations goal scaling deep learning techniques train large dbns approach combines intrinsic parallelism ensemble learning algorithms random dropout approach improve generalization results neural networks using random dropout approach trains separate dbn much smaller original dbn individual graphical processor large cluster combines results using four proposed methods compared existing approaches random approach several noticeable benefits first becomes possible train huge number different networks reasonable time since number parameters trained single machine much smaller original dbn second approach permits better use modern gpu memory due reduced model size third data transferring processors would incur less communication overhead proposed approach train large dbns propose approach supports distributed computation neural networks high level approach consists two steps model parallelism section model combination section first model parallelism step approach automatically parallelizes computation machine using available resources manages communication synchronization data transfer machine second step approach supports four different ways combine results machine form unified dbn model parallelism figure illustrates model parallelism step two machines worthy noting computational needs training dbn one machine depends connectivity structure random dropout significantly reduce complexity dbn well number parameters dropping neurons layer lead reduction parameters connection weights general given dropout probability approach permits model parallelism among machines machine updating disjoint portion weight matrix fundamentally different existing model parallelism approaches example distbelief framework user needs define computation takes place machine layer model approach distributes computation training dbn fully automatically machine addition framework like distbelief complexity dbn reduced rather dbn partitioned graph available machines machine must communicate frequently parameter server updating weights therefore large models high computational demands might processor combining dbns averaging weights majority vote predications synchronized weight update asynchronized weight update processor figure example model parallelism neural network machines example machine randomly drops portion neurons trains separate dbn independently later approach combines trained dbns using four methods described section compute calculate gradient fetch weight queue push machines weight queues figure asynchronous parameter updating algorithm machine training data machine weight queue receiving updated parameters machines beginning machine replica parameters efit access cpus memory beginning limited bottleneck communication costs dominate point contrast approach dbns produced random dropout tend amenable extensive distribution structures given less complex structures lower communication requirements also help alleviate bottleneck many machines waiting single slowest machine finish given phase computation model combination approach provides four ways combine trained dbn machine averaging weights trained dbn machine majority voting predication result test data using trained dbn machine implementation breaks possible cycles arbitrarily ranking predication results occur experiments synchronously updating parameter weights dbn training machine asynchronously updating parameter weights dbn training machine first two ways model combination straightforward omitted paper space reasons next describe update parameter weights synchronously figure sketches asynchronous parameter weight updating algorithm machine training data asynchronous algorithm prevents machine waiting others finish proceeding next training data sacrificing data consistency parameters possible two machines updating parameters simultaneously without explicit ordering consequence asynchronous algorithm may introduce additional stochasticity training compute calculate gradient fetch parameter server wait machines finish updating update parameter server wait machines finish updating figure sychronous parameter updating algorithm machine training data parameter serve stores weights machine fetches needed weights server updating figure sketches synchronous parameter weight updating algorithm machine algorithm central parameter server storing weights parameters end set training data machine sends request parameter server fetch needed parameter weights machines updating requested parameters machine needs wait machines finish updates comparing asynchronous algorithm algorithm eliminates possible data races improves data consistency parameters introduces higher overhead implementation implemented approach prototype using matlab python prototype uses theano library define optimize evaluate mathematical expressions involving multidimensional arrays efficiently specifically implementation uses theano achieve significant speedup calculations leveraging transparent use gpu combining trained dbns implementation uses communication ipc exchange data among multiple threads one processes implementation publicly available http evaluation details dropout training minst dataset architectures deep network number layers number hidden units layer varies different benchmark tasks first step develop prototype evaluate effectiveness mnist handwritten digits dataset consists digit images training testing objective classify digit images correct digit class experimented neural network size pretrained network restricted boltzmann machine rbm epochs one epoch means pass training data phase employs greedy contrastive learning algorithm learning rate exponentially decaying initial value decay rate epoch training weights updated end size momentum also used initial value linear increasing rate first epochs stays fine tuning using dropout epochs employed stochastic gradient descent size use dropout rates hidden units dropout input pixels constant learning rate used constraints imposed norms weight vectors generalization performance function dropout probability original dropout article hinton claims better generalization performance achieved various dropout probabilities implementation details shown section show test error rate varies function dropout probability demonstrated figure dropout decrease test error rate less misclassified examples test data set contrary claim found generation performance dropout actually depends dropout probability dropout probability greater test error rate increases significantly inconsistency might due much smaller training epochs epochs used hinton paper used implementation figure test error rate mnist variety dropout probabilities input visible neurons also use dropout best previous published results task using dbn without dropout rbm without rbm using epochs generalization performance distributed algorithm evaluate generalization performance mnist dataset using various algorithms combine results different machines listed section algorithms error rate sequential weight averaging majority vote synchronous update asynchronous update table test error rate using different algorithms epochs fine tuning weight averaging majority vote algorithms collect final weights independent runs standard dropout algorithms synchronous update asynchronous update algorithms combine results two processes input instance dropout rate algorithms weight averaging synchronous update algorithms achieved notable improvement generalization performance surprisingly majority vote method reduce test error rate large margin asynchronous update algorithm introduced additional noise weight updates lock free mechanism thus generalization performance relatively sequential dropout algorithm however exhibited figure convergence rate synchronous asynchronous update algorithms faster sequential dropout algorithm limitations current evaluation performed desktop dual core intel processor ram nvidia graphics card tuning generally time consumption machine due time constraints able evaluate proposed algorithm relatively small mnist dataset however plan evaluate algorithm speech object recognition benchmark tasks timit continuous speech corpus reuters corpus news article topic recognition imagenet dataset millions labeled images thousands categories figure test error rate mnist function number epochs using updates conclusion paper proposes approach scale deep belief networks dbns core approach use random dropout prevent training data dbn reduce overfitting enable dbn training use computational power clusters distributed environment empirically implementation outperforms approaches promising nearly linear speedups furthermore approach allows parallel training dbn even gradients computationally intensive future work would interesting compare approach approaches using different abstractions example powergraph abstraction exploits internal structure graph programs address challenges computation natural graphs thus may possible adapt similar random dropout idea reduce memory consumption communications processors investigation generalize approach structures problems would enable even faster computation machine learning problems references geoffrey hinton simon osindero teh fast learning algorithm deep belief nets neural july songlin zhao fixed adaptive budget robust kernel adaptive filtering university florida bernhard alex smola support vector machines encyclopedia biostatistics yoav freund robert schapire generalization learning application boosting journal computer system sciences tianqi chen hang qiang yang yong general functional matrix factorization using gradient boosting proceeding international conference machine learning icml volume pages nan wang new boosting algorithm based dual averaging scheme international journal innovative science modern engineering deng dong platt scalable stacking learning building deep architectures acoustics speech signal processing icassp ieee international conference pages march dan claudiu ciresan ueli meier luca maria gambardella schmidhuber deep big simple neural nets excel handwritten digit recognition corr jia raymond tse linda shapiro learning rank severity unrepaired cleft lip nasal deformity mesh data pattern recognition icpr international conference pages ieee yoshua bengio ducharme pascal vincent christian janvin neural probabilistic language model mach learn march quoc jiquan ngiam adam coates ahbik lahiri bobby prochnow andrew optimization methods deep learning lise getoor tobias scheffer editors proceedings international conference machine learning icml pages new york usa june acm rajat raina anand madhavan andrew deep unsupervised learning using graphics processors proceedings annual international conference machine learning icml pages new york usa acm geoffrey hinton nitish srivastava alex krizhevsky ilya sutskever ruslan salakhutdinov improving neural networks preventing feature detectors corr pages john langford smola martin zinkevich slow learners fast nips pages gideon mann ryan mcdonald mehryar mohri nathan silberman daniel walker efficient distributed training conditional maximum entropy models advances neural information processing systems ryan mcdonald keith hall gideon mann distributed training strategies structured perceptron human language technologies annual conference north american chapter association computational linguistics hlt pages stroudsburg usa association computational linguistics martin zinkevich markus weimer alex smola lihong parallelized stochastic gradient descent lafferty williams zemel culotta editors advances neural information processing systems pages feng niu benjamin recht christopher stephen wright hogwild approach parallelizing stochastic gradient descent nips pages jeffrey dean sanjay ghemawat mapreduce simplified data processing large clusters commun acm january joseph gonzalez yucheng low haijie danny bickson carlos guestrin powergraph distributed computation natural graphs proceedings usenix symposium operating systems design implementation osdi hollywood october george dahl student member dong senior member deng alex acero deep neural networks large vocabulary speech recognition ieee transactions audio speech language processing deng dahl mohamed jaitly senior vanhoucke nguyen sainath kingsbury deep neural networks acoustic modeling speech recognition ieee signal processing magazine jeffrey dean greg corrado rajat monga kai chen matthieu devin quoc mark mao marc aurelio ranzato andrew senior paul tucker yang andrew large scale distributed deep networks advances neural information processing systems theano library http richard stevens bill fenner andrew rudoff unix network programming vol pearson education edition mohamed dahl hinton acoustic modeling using deep belief networks ieee transactions audio speech language processing tony rose mark stevenson miles whitehead reuters corpus volume yesterdays news tomorrows language resources proceedings third international conference language resources evaluation pages jia deng wei dong richard socher jia kai imagenet hierarchical image database cvpr aapo kyrola guy blelloch carlos guestrin graphchi graph computation proceedings usenix symposium operating systems design implementation osdi hollywood october
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sep reachability analysis nonlinear systems using componentwise contraction properties murat arcak john maidens department electrical engineering computer sciences university california berkeley september abstract shortcoming existing reachability approaches nonlinear systems poor scalability number continuous state variables mitigate problem present simulationbased approach first sample number trajectories system next establish bounds convergence divergence samples neighboring trajectories compute bounds using contraction theory reduce conservatism partitioning state vector several components analyzing contraction properties separately direction among benefits allows analyze effect constant uncertain parameters treating state variables partitioning separate direction next present numerical procedure search weighted norms yield prescribed contraction rate incorporated reachability algorithm adjust weights minimize growth reachable set introduction reachability analysis critical testing verification control systems formal control synthesis reachability dictates transitions abstraction system continuous dynamics existing reachability approaches nonlinear systems include level set methods linear piecewise linear approximations nonlinear models followed linear reachability techniques interval taylor series methods differential inequality methods however results typically scale poorly number continuous state variables limiting applicability practice hand approaches scale well state dimension take advantage inexpensive numerical simulations naturally parallelizable leveraged concepts contraction theory develop new approach first sample number trajectories system next establish bounds divergence samples neighboring trajectories use bounds provide guaranteed reachable set unlike uses lipschitz constants bound divergence trajectories use matrix measures take negative values thus allowing convergence trajectories reducing conservatism another related reference uses sensitivity equations track convergence divergence properties along simulated trajectories however approach guarantee computed approximation contains true reachable set note generalize partitioning state vector several components analyzing growth contraction properties direction defined component unlike searches single growth contraction rate cover every direction state space new approach takes advantage directions offer favorable rates generalization analyze effect constant uncertain parameters treating state variables partitioning separate direction along growth occurs related approach employed every state variable defines separate direction however may lead overly conservative results since dynamics associated multiple state variables may possess favorable rate individual state variables isolation section present main contraction result corollary serves starting point reachability algorithm section detail algorithm demonstrate example significantly reduce conservatism section present numerical procedure search weighted norms yield prescribed contraction rate incorporated reachability algorithm adjust weights minimize growth reachable set propagates time sequel make use matrix measures defined let norm let denote induced matrix norm measure matrix upper derivative direction lim hak unlike norm matrix measure take negative values evident table vector norm maxj induced matrix norm maxj maxj maxi induced matrix measure maxj ajj maxj maxi aii table commonly used vector norms corresponding matrix norms measures componentwise contraction consider nonlinear dynamical system continuous continuously differentiable partition state vector components xtk rni likewise decompose jacobian matrix conformal blocks jij rni following proposition gives growth bound two trajectories system variants proposition appear provide independent proof section proposition let constant matrix jii cij kjij domain two trajectories every trajectory starting line segment remains time exp denotes inequality use different vector norm component say provided interpret jii cij kjij kpi matrix measure kpi mixed norm defined kakpi max next derive corollary proposition useful reachability analysis let denote solution starting define reachable set time initial set reacht likewise define reachable set time interval reach reacht corollary let trajectory define norm ball initial conditions centered suppose coarse set available reach satisfies reacht exp corollary relies coarse reachable set find constant matrix satisfying uses find accurate reachable set end time interval one choose bounded invariant set system entire state space global upper bound exists side less conservative estimate one find bound component vector field invariant set interest let gives tighter bound interval length smaller reachability algorithm given sequence simulation points algorithm tracks evolution initial norm ball along trajectory applying corollary along bound interval algorithm algorithm bounding reachable tube along sample trajectory require vector initial ball size sequence simulation points bounds set compute matrix satisfies set exp end return similar approach reachability pursued using special case proposition choice amounts looking single growth contraction rate cover every direction state space conservative directions offer favorable rates others extreme used also lead overly conservative results since dynamics associated multiple state variables may possess favorable rate individual state variables isolation following example illustrates intermediate choices may give tighter bounds extremes example consider harmonic oscillator treat constant frequency state variable satisfying view different values variations initial condition thus state vector jacobian matrix partition components matrix measures norms induced euclidean norm thus satisfies invariant set follows corollary initial norm ball evolves nominal frequency sample trajectory obtained note correctly predicts absence growth direction accounting effect frequency variation enlarging radius corresponding ball algorithm gives tighter estimate applying corollary along figure algorithm applied sample trajectory initial norm ball plot left takes means uncertainty around nominal frequency uncertainty leads growth radius norm ball direction plot right takes since uncertainty frequency radius norm ball direction remains constant bound used algorithm calculated invariant set bound every interval simulated trajectory result algorithm shown figure left uncertainty allowed around nominal frequency right panel shows result case uncertainty frequency radius norm ball remains constant example applied proposition partitioning state components alternative choice partition amounts searching single growth rate direction fails identify lack change direction indeed matrix measure positive choice norm thus norm ball grows every direction choice also overly conservative misses property combined dynamics instead applying matrix form falsely predicts rapid growth norm ball direction even uncertainty present frequency automatic selection weighted norms demonstrated bounding matrix measure induced weighted expressed constraints convex functions weights used fact develop heuristic minimizing growth reachable set propagates time authors use linear matrix inequality lmi constraint corresponding weighted together interval bounds jacobian argue optimal bounds jacobian matrix measure computed automatically solving sequence semidefinite programs sdps section demonstrate types methods extended case componentwise contraction proposition show fixed matrix set weight matrices weighted version holds convex set expressed conjunction infinite number lmis proposition show polytopic bounds jacobian exploited compute inner approximation set feasible weight matrices enabling weight matrices selected automatically using standard numerical sdp solver begin two lemmas demonstrate weighted matrix measure norm bounds expressed lmis throughout section inequality symbol used denote positive semidefinite matrix lemma lemma positive definite matrix norm lemma positive definite matrices kak denotes mixed norm kak max proof implies thus max equivalently kak combining lemmas along reasoning used derive equation arrive following result proposition given matrix search weighted euclidean norms satisfied formulated semidefinite program find subject jii jii jij jij pit note contains infinite number lmi constraints therefore solved numerically using standard sdp solver address show conservative inner approximation feasible set defined terms finite conjunction lmis stating result prove following lemma shows infinite family lmis conservatively approximated finite family lmis assuming existence polytopic bounds coefficient matrices lemma let family symmetric matrices parameterized assume exist finite set matrices fiki family bounded matrix polytope vertices fiki conv fiki conv denotes convex hull fnkn implies proof let using assumption know exist nonnegative weights fiki therefore fnkn fnkn fnkn state result allows find set weights satisfying solving finite set lmis proposition let basis space symmetric matrices suppose exist matrices jii jii conv jij jij conv kij kij nij solution sdp find subject kij ejkj kij nij yields solution proof follows straightforward manner expanding decision variables applying lemma note compact continuous always possible find collection matrices proof proposition let denote solution initial condition particular taking derivative sides respect get means variable satisfies conclude lemma jii kjij denotes upper derivative respect since holds conclude cii cij means since matrix metzler cij theorems positive systems follows standard comparison exp note thus substituting get exp next note implies noting exp combining obtain lemma consider linear system continuous suppose decompose blocks aij rni let rni constitute conformal partition lim aii kaij proof lemma note lim lim lim haii aij haii lim aij lim haii follows definition matrix norm kaij references kapinski deshmukh jin ito butts approaches verification embedded control systems overview traditional advanced modeling testing verification techniques ieee control systems dec tabuada verification control hybrid systems symbolic approach springer mitchell bayen tomlin formulation reachable sets continuous dynamic games ieee trans autom control althoff stursberg buss reachability analysis nonlinear systems uncertain parameters using conservative linearization ieee conf decision control pages chutinan krogh computational techniques hybrid system verification ieee trans autom control youdong lin mark stadtherr validated solutions initial value problems parametric odes appl numer neher jackson nedialkov taylor model based integration odes siam numer anal lakshmikantham leela differential integral inequalities volume academic press new york joseph scott paul barton bounds reachable sets nonlinear control systems automatica alexandre oded maler systematic simulation using sensitivity analysis hybrid sys comp control pages zhenqi huang sayan mitra computing bounded reach sets sampled simulation traces hybrid sys comp control pages agung julius george pappas trajectory based verification using local invariance hybrid sys comp control pages springer john maidens murat arcak reachability analysis nonlinear systems using matrix measures ieee transactions automatic control lohmiller slotine contraction analysis nonlinear systems eduardo sontag contractive systems inputs perspectives mathematical system theory control signal processing pages matthias rungger majid zamani scots tool synthesis symbolic controllers proceedings international conference hybrid systems computation control hscc desoer vidyasagar feedback systems properties society industrial applied mathematics philadelphia originally published academic press new york kapela algorithm control systems ordinary differential equations discrete continuous dynamical systems series russo bernardo sontag contraction approach hierarchical analysis design networked systems ieee transactions automatic control reissig weber rungger feedback refinement relations synthesis symbolic controllers ieee transactions automatic control chuchu fan james kapinski xiaoqing jin sayan mitra locally optimal reach set overapproximation nonlinear systems proceedings international conference embedded software emsoft pages aminzare shafi arcak sontag guaranteeing spatial uniformity reactiondiffusion systems using weighted contractions kulkarni stan raman editors systems theoretic approach systems synthetic biology models system characterizations pages
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algebraic invariants projective monomial curves associated generalized arithmetic sequences dec isabel bermejo eva ignacio bstract let infinite field let generalized arithmetic sequence positive integers exist consider projective monomial curve pnk parametrically defined tmn tmn smn tmn work characterize koszul properties homogeneous coordinate ring whenever also obtain formula type moreover divides obtain minimal basis vanishing ideal respect degree reverse lexicographic order derive formulas regularity hilbert series hilbert function terms sequence ntroduction let polynomial ring variables infinite field let positive integers consider projective monomial curve pnk parametrically defined tmn tmn smn tmn consider homomorphism induced smi tmn smn tmn since infinite prime ideal ker vanishing ideal see corollary thus homogenous coordinate ring moreover homogeneous binomial ideal see lemmas regularity regularity defined follows minimal graded free resolution reg max eij see equivalent definitions whenever integer type addition gorenstein corresponding ideal complete intersection generated set polynomials said koszul algebra residue class field linear free resolution study koszul algebras topic considerable current interest recent account theory refer reader denote space spanned homogeneous polynomials degree hilbert function hilbert series hfk dimk hfk respectively paper study already mentioned invariants properties generalized arithmetic sequence relatively prime integers exist gcd context prove arithmetic sequence three authors partially supported ministerio competitividad spain isabel bermejo eva ignacio remarkable result obtained consequence new result providing large family projective monomial curves arithmetically also provide alternative proof theorem determine complete intersection either even moreover prove koszul consecutive numbers addition study detail generalized arithmetic sequence divides setting provide formula reg end main reference paper authors prove projective monomial curve regularity equal regularity denotes initial ideal respect degree reverse lexicographic order see section moreover obtain regularity maximum regularities irreducible components see corollary since regularity irreducible monomial ideal easy compute strategy consists obtaining minimal basis respect degree reverse lexicographic order get irreducible components allow deduce value reg terms arithmetic sequence obtaining result separate two cases case arithmetic sequence section non one divides section also exploit knowledge describe hilbert series hilbert function additionally case provide formula type yielding characterization gorenstein property results case extend natural way consider particular setting form minimal set generators semigroup generate contents paper following section devoted arithmetic sequences key result theorem provides byproduct derive corollary homogeneous coordinate ring projective monomial curve associated arithmetic sequence proposition gives irreducible components consequence theorem get value regularity terms sequence theorems provide hilbert series hilbert function type addition give corollary characterization gorenstein property third section begin providing theorem large family projective monomial curves whose corresponding coordinate ring precisely prove whenever projective monomial curve associated sequence integers containing generalized arithmetic sequence least terms satisfying additional hypotheses rest section concerns generalized arithmetic sequence particular byproduct theorem derive corollary setting unless arithmetic sequence consequence characterize corollary complete intersection property study detail setting divides hypotheses obtain basis theorem also minimal set generators proposition present irreducible decomposition formula reg provided theorem corollary prove regularity always attained last step minimal graded free resolution hilbert series hilbert function hilbert polynomial given theorems corollary respectively last section devoted study koszul property particular theorem characterizes koszul algebra generalized arithmetic alg invariants projective mon curves assoc generalized arithmetic sequences sequence finally also characterize koszul property theorem theorem rithmetic sequences section concerns study arithmetic sequence relatively prime integers key point section prove theorem describe minimal basis respect degree reverse lexicographic order begin technical lemma associate sequence pair pair integers plays important role rest paper lemma take set qmn forpall min nmi proof trivial let prove set min nmi follows qmn assume contradiction proving following facts clearly lead contradiction iii divides gives get also iii gcd assumption obtain mmn mmn proves completes proof following notation lemma state main result section theorem xqn minimal basis respect degree reverse lexicographic order proof one easily check using lemma let prove hin contradiction suppose hin claim exist two monomials hin indeed hin cosets monomial hin form hin exist monomials since exists suppose hin also assume gcd prime since may assume xjj isabel bermejo eva ignacio otherwise hin moreover since get may assume gcd however setting implies hence contradiction assume firstly observe otherwise contradiction hence get consider several different cases case becomes lemma get contradiction case becomes lemma get contradiction case subcase one derive lemma implies contradiction subcase one easily obtain implies thus contradiction moreover one easily check minimal observe theorem variables appear minimal set generators thus apply proposition deduce following result corollary worth mentioning result generalization proposition authors obtain result restrictive setting minimal system generators semigroup nmi worth mentioning techniques different remark basis provided theorem also minimal set generators see remark thus first betti number minimal graded free resolution equals next objective provide explicit description irredundant irreducible decomposition initial ideal respect degree reverse lexicographic order ideal irreducible written intersection two ideals properly containing monomial ideal irreducible generated pure powers variables see proposition moreover irreducible decomposition monomial ideal expression intersection irreducible monomial ideals redundant ideal intersection expression unique see theorem call irredundant irreducible decomposition proposition let lemma irredundant irreducible decomposition alg invariants projective mon curves assoc generalized arithmetic sequences irredundant irreducible decomposition proof let denote every denote otherwise aim proving theorem minimal system generators respect degree reverse lexicographic order observe thus order see reverse inclusion take monomial three possibilities xbn xbn xbn holds observe thus remains see redundant ideal intersection observe hence hence redundant redundant claimed introduction reg reg regularity expressed terms regularity irreducible components due fact monomial ideal nested type associated primes form various following result states relation lemma corollary let monomial ideal nested type let irredundant irreducible decomposition reg max reg moreover irreducible monomial ideal whose monomial minimal generators degrees known reg see remark following result gives value regularity terms arithmetic sequence theorem let arithmetic sequence gcd reg proof take lemma direct application proposition lemma yields reg mod hence moreover exploit description provide explicit description hilbert series hilbert function type results also obtained inp restrictive setting minimal system generators nmi going particular case arithmetic isabel bermejo eva ignacio sequence introduce general result expressing hilbert series terms monomials belonging result applies projective monomial curve proposition let sequence positive integers hilbert series moreover proof well known hilbert series coincides observe since considering degree reverse lexicographic order homogeneous prime ideal appear minimal generator thus follows hence get moreover conclude result suffices observe additionally proposition result follows consequence arithmetic sequence relatively prime integers obtain following result theorem hilbert series proof since corollary apply theorem get hxn deduce result alg invariants projective mon curves assoc generalized arithmetic sequences remark worth pointing theorem also get value reg indeed regularity attained last step minimal graded free resolution see proposition moreover hilbert series related minimal graded free resolution formula eij thus reg equals degree since reg reg theorem hilbert function hfk proof hilbert function coincides thus theorem hfk hfk finally observe lemma remark since regularity one unit bigger regularity hilbert function see theorem regularity hilbert function smallest integer hilbert polynomial hilbert function coincide thus using approach theorem allows reprove theorem hilbert polynomial hfk moreover easy check hfk nevertheless hfk thus regularity hilbert function equals obtain type theorem take type proof set consider grading induced deg deg also consider sets deg theorem type number elements theorem isabel bermejo eva ignacio satisfies mod claim hence moreover easy check claim follows easily therefore type otherwise thus result follows definition theorem easily deduce following characterization gorenstein property corollary gorenstein mod let illustrate results section examples example consider set observe theorem get reg since theorem obtain type hence gorenstein moreover theorem hilbert series theorem hilbert function hfk regularity hilbert function see remark example consider set observe reg since type moreover hilbert series hilbert function respectively hfk regularity hilbert function eneralized arithmetic sequences section concerns study generalized arithmetic sequence discard case two integers always form arithmetic sequence thus study covered previous section first goal section prove whenever generalized arithmetic sequence prove direct consequence theorem general result providing big family projective monomial curves arithmetically concretely result concerns projective monomial curves defined sequence containing generalized arithmetic alg invariants projective mon curves assoc generalized arithmetic sequences sequence terms finishing exist mij theorem let contain generalized arithmetic sequence mil proof proving exists monomial minimal generator involving variable thus proposition conclude consider binomial observe exists minimal generator involving assume exists claim indeed degree xhn implies contradiction consider binomial since get nevertheless exists minimal generator involving result follows theorems directly derive following characterization cohenmacaulay property generalized arithmetic sequence corollary arithmetic sequence moreover consequence corollary remark easily recover theorem complete intersection property characterized corollary let generalized arithmetic sequence relatively prime integers complete intersection even proof assume complete intersection corollary arithmetic sequence hence remark complete intersection equality hold equivalently even direct application remark gives result rest section assume generalized arithmetic sequence relatively prime integers divides turns hypotheses closely related projective monomial curve associated arithmetic sequence equivalently arithmetic sequence relatively prime integers studied detail previous section sake convenience assume following result shows initial ideals respect degree reverse lexicographic order related useful rest section proposition proof obvious prove converse consider assume without loss generality exists consider two cases case case proving exists andpg firstly claim indeed gcd divides isabel bermejo eva ignacio thus moreover exist otherwise implies since homogeneous contradiction take consider one easily check iterating argument necessary get result remark divide easy find examples conclusion proposition hold example consider set proposition however help computer one check even consider gcd observe order describe minimal basis introduce two technical lemmas lemma take set pmn min nmi proof straightforward check let denote min nmi observe divides divides gcd hence min nmi applying lemma get proved following notation previous lemma state next result lemma set min nmi exist unique setting following proof definition write suppose exist assume without loss generality get hence get let prove first consider case since hand implies derive thus finally consider case equalities obtain deduce alg invariants projective mon curves assoc generalized arithmetic sequences hand obtain suppose get contradiction conclude remark lemma gives algorithmic way computing integers starting given lemma nevertheless one easily derive explicit formula indeed integer mod remark notice min nmi applying lemma sequence relatively prime integers get moreover denotes projective monomial curve associated arithmetic sequence reg equals following notation lemmas describe minimal basis respect degree reverse lexicographic order theorem let minimal basis respect degree reverse lexicographic order projective monomial curve associated arithmetic sequence minimal basis respect degree reverse lexicographic order moreover minimal set generators proof set xnj lemma moreover since clearly let prove ideal generated initial terms binomials respect degree reverse lexicographic order consider binomial gcd consider different cases case proposition get case divides divides gcd exists divides deduce remains consider divides lemma taking definition thus divides proof complete prove minimal set generators suffices prove basis described theorem remark minimal set generators since elements degree belong follows elements degree redundant let take degree divided monomials appearing elements hence isabel bermejo eva ignacio redundant assume deg deg deg consider homomorphism induced applying derive exists linear form binomial prime ideal hxj hence contradiction thus minimal set generators remark putting together theorems obtain minimal basis respect degree reverse lexicographic order observe consider basis set obtained minimal basis remark remove assumption divides general minimal basis respect degree reverse lexicographic order may minimal system generators indeed consider first example remark one minimal basis elements whereas first betti number next result exploit description minimal basis obtained theorem get irreducible decomposition proposition let lemmas let irreducible components irredundant irreducible decomposition proof set hxjh theorem obtain hxjh minimal system generators respect degree reverse lexicographic order easy check order see reverse inclusion consider distinguish two possibilities setting consequence thus holds remains show redundant ideal intersection argument theorem applies derive redundant finally redundant irreducible decomposition obtain value regularity theorem let generalized arithmetic sequence gcd divides set divide reg divides alg invariants projective mon curves assoc generalized arithmetic sequences proof observe proposition irreducible components direct application lemma yields reg max reg max moreover clear since thus get reg max objective prove maximum equals equivalently reg proof use following three facts theorem remark reg lemma iii since divides nmi lemma divide proof four cases reg reg holds iii therefore get reg conclude holds therefore cases get reg conclude let prove possible contradiction assume holds since reg theorem get claim indeed setting observe nevertheless hence lemma two consecutive values conclude however implies reg see remark contradiction conclude result suffices observe lemma divides following result make use result prove setting regularity always attained last step minimal graded free resolution corollary reg attained last step minimal graded free resolution proof let set according corollary sufficient condition regularity attained last step resolution reg max theorem get xih isabel bermejo eva ignacio projective monomial curve associated arithmetic sequence theorem obtain since conclude max equals reg result follows last part section provide formulas hilbert series hilbert function setting theorem hilbert series hilbert series projective monomial curve associated arithmetic sequence proof proposition setting observe theorem unique minimal generator appears derive derive last equality directly deduced theorem done direct consequence previous result obtain simpler expression hilbert series corollary hilbert series odd even odd even alg invariants projective mon curves assoc generalized arithmetic sequences theorem hilbert function hfk hfk projective monomial curve associated arithmetic sequence hfk whenever proof hilbert function coincides thus hfk deg write hfk deg deg theorem easily deduce deg deg deg thus hfk hfk deg moreover theorem get deg completes proof moreover also describe hilbert polynomial terms hilbert polynomial described theorem see also remark corollary hilbert polynomial get result suffices observe proof set theorem let illustrate results section example example consider set observe relatively prime divides compute observe divide theorem get reg moreover expression get following remark get theorem obtain observe sequence defines projective monomial curve sequence example get hence conclude equals isabel bermejo eva ignacio koszulness section characterize koszul algebra generalized arithmetic sequence theorem theorem theorem recall see following well known implications iii hold possesses quadratic basis koszul iii generated quadrics implications frequently used section theorem let generalized arithmetic sequence relatively prime integers koszul algebra consecutive numbers proving result make use following lemma lemma take generated quadrics proof set minimum integer turns exists minimal set generators containing see lemma hence quadric implies thus conclude otherwise proof theorem since generated quadrics lemma get thus moreover lemma derive observe arithmetic sequence consequence basis theorem consists quadrics conclude koszul algebra rest section aim listing projective monomial curves corresponding algebra koszul theorem theorem theorem gcd koszul algebra theorem gcd koszul algebra one following sequences obtain results first introduce two lemmas concerning new projective curve defined precisely projective monomial curve parametrically defined tmn tmn smn tmn tmn denote vanishing ideal denote integer gcd gcd lemma take set koszul algebra moreover also koszul algebra alg invariants projective mon curves assoc generalized arithmetic sequences proof thus theorem corollary complete intersection generated quadrics implies koszul see corollary theorem observe therefore lemma proposition get also complete intersection generated quadrics koszul algebra lemma set gcd consecutive numbers koszul algebra proof theorem proof basis respect degree reverse lexicographic order provided theorem consists quadrics moreover observe thus lemma proposition get let prove basis respect monomial ordering given degree reverse lexicographic order indeed take suppose consider three different cases case exist case thus exists follows zmi implies contradiction case thus quadratic basis respect therefore koszul algebra proof theorem since generated quadrics lemma get either since gcd easily derive equals koszul lemma equals conclude koszul theorem proof theorem since generated quadrics lemma case case analysis possible values together fact gcd proves implication get koszul algebra theorem lemma get koszul sets moreover prove implication applying lemma finally direct computation isabel bermejo eva ignacio yields basis respect degree reverse lexicographic order formed quadrics koszul well acknowledgments authors partially supported ministerio competitividad spain second author also partially supported cajacanarias eferences bayer mumford computed algebraic geometry computational algebraic geometry commutative algebra cortona cambridge univ press cambridge bermejo complete intersections simplicial toric varieties symbolic comput part bermejo complete intersections certain affine projective monomial curves bull braz math soc new series bermejo gimenez regularity projective curves proc amer math soc bermejo gimenez computing regularity subeschemes pnk using quotients monomial ideals pure appl algebra bermejo gimenez saturation regularity algebra cavaliere niesi monomial curves type manuscripta math conca negri rossi koszul algebras regularity commutative algebra springer new york eisenbud goto linear free resolutions minimal multiplicities algebra patil roberts bases ideal generators projective monomial curves comm algebra molinelli tamone hilbert function certain rings monomial curves pure appl algebra sturmfels bases convex polytopes university lecture series american mathematical society providence villarreal monomial algebras monographs textbooks pure applied mathematics marcel dekker new york villarreal monomial algebras second edition monographs research notes mathematics chapman facultad iencias niversidad aguna aguna pain address ibermejo facultad iencias niversidad aguna aguna pain address evgarcia lip ens lyon cnrs inria lyon umr lyon rance address iggarcia
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performance test supermartingale confidence intervals success probability bernoulli trials peter emanuel kevin yanbao department sep applied mathematics university colorado boulder boulder colorado usa national institute standards technology boulder colorado usa center theory quantum matter university colorado boulder colorado usa institute quantum computing department physics astronomy university waterloo waterloo ontario canada ntt basic research laboratories ntt corporation atsugi kanagawa japan given composite null hypothesis test supermartingales supermartingales respect initial value large values test supermartingales provide evidence result test supermartingales effective tool rejecting particularly obtained small serve certificates null hypothesis examples include rejection local realism explanation bell test experiments foundations physics certification entanglement quantum information science test supermartingales advantage adaptable experiment allowing arbitrary stopping rules inversion acceptance regions also used determine confidence sets use example compare performance test supermartingales computing confidence intervals bounds exact example problem inferring probability success sequence bernoulli trials cost using technique restriction stopping rules particular test supermartingale study quantifies cost contents introduction basic concepts iii bernoulli hypothesis tests comparison comparison confidence intervals corresponding author conclusion vii appendix preliminaries expression ppbr approximations asymptotic normality log differences confidence interval endpoints acknowledgments references figures introduction experiments physics require high confidence justify claims discovery unambiguously exclude alternative explanations particularly striking examples foundations physics experiments demonstrate theories based local hidden variables called local realist theories explain statistics observed quantum experiments called bell tests see ref review refs definitive experiments date successful bell tests imply presence randomness observed statistics result one notable applications bell tests randomness generation application necessary certify randomness generated certificates equivalent extremely small significance levels appropriately formulated hypothesis test general extreme significance levels frequently required protocols communication computation ensure performance bell tests consist sequence trials gives result models restrict statistics therefore constitute composite null hypothesis rejected traditionally data analyzed estimating value bell function standard error collective result statistics see null hypothesis expected negative large value compared considered strong evidence null hypothesis method suffers several problems including failure gaussian approximation extreme tails fact trials observably independent identically distributed ref method introduced give rigorous bounds bounds without assumptions dependence trial statistics earlier trials method seen application test supermartingales defined ref test supermartingales first considered many basic properties proved ville work introduced notion martingales method involves constructing stochastic process determined initial value models expectations conditional past events explained final value sequence trials expectation bounded inverse bound according markov inequality large observed value test supermartingale thus provides evidence models refs give methods construct achieve asymptotically optimal gain rate log trials expectation functional typically improvement valid bell tests additional benefits constructed adaptively based observed statistics bounds remain valid even experiment stopped based current value techniques successfully applied experimental data bell test photons methods fail although terminology apparently relatively recent test supermartingales traditionally played major theoretical role carefully constructed test supermartingales contribute asymptotic analysis distributions proofs large deviation bounds constructed null hypothesis viewed set distributions used adaptive hypothesis tests generality application bell tests shows least regime high significance results required test supermartingales perform well better methods compare performance test supermartingales directly standard large deviation bounds based inequality exact calculations comparison case calculations performed efficiently namely testing success probability bernoulli trials three bounds thus obtained asymptotically optimal gain rates surprisingly given experiment test supermartingales yield systematically worse bounds difference much smaller variation effect viewed cost robustness arbitrary stopping rules ease calculation use optimal test supermartingale construction expect similar results matter test supermartingale used hypothesis test parametrized used construct confidence regions acceptance region inversion see ref sect motivated observation consider use test supermartingales determining confidence regions expect perform well regime increase region size associated robustness stopping rules therefore compared methods mentioned determining confidence intervals success probability bernoulli trials normalizing difference interval endpoints success probability standard deviation find large deviation bounds exact regions differ constant fixed confidence levels test supermartingale normalized endpoint deviation log instead effect noted ref partially reflects suboptimalpchoice supermartingale maintain robustness stopping rule one expects log log according law iterated logarithm however note number trials fixed advance normalized endpoint deviation reduced adaptive test supermartingale although increase confidence region necessitated stopping rule robustness large reasonably sized known ahead time principle avoided without losing ability adapt test supermartingale fly experiment situations remainder paper structured follows establish notation used define basic concepts sect also explain adaptivity help reject hypotheses stochastic processes introduce three methods applied bernoulli trials sect iii also establish basic monotonicity properties relationships three bounds obtained sect determine behavior bounds detail including asymptotic behavior sect introduce confidence intervals obtained acceptance region inversion focus intervals determined lower bounds note results apply intervals observations sects based theorems whose proofs found appendix many observations sections ignore asymptotically small terms results appendix uncompromisingly determine interval bounds relevant expressions explicit constants concluding remarks found sect basic concepts use usual conventions random variables rvs values rvs denoted capital letters values corresponding lower case letters rvs finite valued probabilities expectations denoted respectively formula expression refers event formula true notation refers distribution induced space values use usual conventions conditional probabilities expectations also denotes probability distribution induced conditional event consider stochastic sequences rvs think outcomes sequence trials study consider rvs standard parameter bernoulli satisfying parameter also referred success probability denote distribution define extend conventions greek letter value determined values may omit subscripts statistics based full set samples expressions involving require integer assured definition null hypothesis equivalent set distributions refer null study bernoulli rvs consider nulls parametrized set distributions bernoulli rvs one test null hypothesis determined null means special statistics called bounds statistic bound subscript indicates distribution respect probabilities calculated usually write instead bound even bounds achieved member small strong evidence null since interested small pvalues preferentially use negative logarithm log call log work logarithms base default general method constructing start arbitrary jointly distributed usually function define tail probability argument standard define function need show since set either form qmin qmin qmin first case qmin qmin second qmin limn qmin limn qmin applied countable monotone union referring null mean consists distributions distributed according fixed independent beyond extend set distributions property depends denote extended null particular models mentioned introduction constitute particular null hlr sequences trials called bell tests ref technique called ratio pbr method introduced construct achieve asymptotically optimal gain rates defined log method best understood way constructing test supermartingale hlr test supermartingale stochastic sequence function work avoid unwanted boundary cases require positive definition test supermartingale used general one consider discrete time avoid customary increasing sequence making dependent explicit stochastic sequence every test supermartingale defines follows one characteristic properties supermartingales markov inequality statistics according martingale theory stopped process stopping rule respect also defines also defines see ref discussion examples test supermartingale viewed running product call test factors defining properties equivalent distributions null pbr method adaptively constructs function next trial outcome earlier trial outcomes method designed testing closed convex null asymptotically optimally gain rates achieved trials trial distribution known optimal test factor would given distribution closest divergence log since known pbr method obtains empirical estimate information available trial determines test factor given test factors satisfy see ref proof applications problem testing ability choose test factors adaptively helps reject extended nulls distributions vary experiment progresses distributions still independent parameters vary parameters depend past outcomes suppose distributions sufficiently stable empirical frequencies past trials statistically close next trial probability distribution tively compute test factor used next trial past trials empirical frequencies example following strategy outlined previous paragraph procedure opportunity reject extended null provided sufficiently long period original null hold example consider extended null true success probabilities trial may vary maybe result changes experimental parameters need calibrated suppose goal calibrate use adaptive test factors find point reject according running product test factors recalibrate experiment recalibration succeeds pushing remaining trials may still reject extended null end experiment many cases analysis performed experiment may possible stop experiment recalibration situation frequencies run trials clearly show adaptive test factors chosen would tend trivial equal case next trials contribute final test factor product contrast hypothesis test based final sum outcomes trials contribute equally let parameter distributions need determine distributions close relationship methods determining confidence sets hypothesis tests let null distributions parameter given family hypothesis tests construct confidence sets inverting acceptance region see ref sect according construction confidence set level given random quantity defining property level confidence set coverage probability satisfies distributions use construction sequences bernoulli rvs null obtain confidence intervals form confidence set interval type refer confidence lower bound endpoint distribution necessarily define use acceptance region inversion extended null obtain confidence region note whose value covered confidence set probability least confidence set need interval general including everything infimum supremum increases coverage probability set converted interval desired focus confidence intervals observations immediately apply intervals ones standard method obtaining confidence interval two intervals example obtain confidence upper bounds level symmetry example relabeling bernoulli outcomes obtain interval level compute lower upper bounds level interval interval bounds coverage probability interval valid according union bound applied maximum probabilities two intervals iii bernoulli hypothesis tests compare three hypothesis tests nulls extended nulls exact test test pch pbr test pvalue ppbr discussing properties tests respect hypothesis parameter true success probability empirical success probability generally assume parameters interior range particular discussing purely functional properties respect values use variable instead default positive integer exact test obtained tail bernoulli rvs defined sect note unlike consider bound achieved member null quantity decreasing function given smooth monotonically increasing function given see compute positive probability given distributed bounded case null restricted distributions follows standard construction pvalues null tails statistics explained previous section extended null follows observations tail probabilities linear functions distribution parameters extremal distributions independent tail probabilities monotonically increasing separately see also ref app define max test optimal bound binary random variable given pch max otherwise setting pch see ref log pch nkl abbreviate pch monotonically increasing decreasing constant pbr test use comparison constructed point null defined ppbr pbr test ppbr max ppbr ppbr shown function ppbr isolated maximum seen differentiating log log log thus maximum achieved therefore write ppbr ppbr otherwise ppbr definition ppbr strictly increasing function strictly decreasing see consider compute ratio successive values follows ppbr final value test supermartingale obtained conthe expression ppbr structing test factors define would empirical estimate two initial trials thus values respectively test factors given one verify generally set compute designed test supermartingale point null thm app vii establishes definition ppbr maximum ensures ppbr show ppbr establish ppbr pch direct calculation ppbr expression maximized seen considering ratios successive values calculation applied also therefore better choice test factors construct test supermartingale test would otherwise choice ensures final value test supermartingale obtained multiplying test factors determined would complicate study summarize observations three tests following theorem theorem pch ppbr three tests satisfy following monotonicity properties strictly increasing strictly decreasing function pch strictly increasing constant strictly decreasing constant ppbr strictly increasing constant strictly decreasing comparison begin determining relationships pch ppbr precisely since interested small convenient focus log instead determine differences identity log pch nkl reference log log pch examine differences determined following theorem theorem log ppbr log pch log log log log pch log log theorem follows thms cor proven appendix explicit interval expressions obtained log differences order notation assumes fixed bounds uniform see expressions appendix details notable observation systematic gaps log log already knew question exact test best three simple application gaps may seem large absolute scale representing factors close fact much smaller variation determine variation consider asymptotic distributions readily determine log asymptotically normal standard deviations proportional transferred variance compared standard deviations gaps negligible next theorem determines specific way asymptotic normality holds let denote normal distribution mean variance notation means converges distribution normal distribution mean variance theorem assume pch ppbr log log converges distribution according log log theorem proven appendix see thm rest paper write one tests matter one display behavior described theorems fig conclude phenomena discussed already apparent small numbers trials fig computed quantiles log numerically using formulas provided previous section substituting corresponding quantile given explicit let defined minimum value satisfying simplicity place quantile middle relevant gap distribution example median monotonicity properties tests log given log noted gaps log form log fact possible determine asymptotic behavior gaps accounting explicitly given terms thm asymptotically normal variances order standard deviations gaps therefore small compared size precise statement asymptotic normality thm appendix fig shows gaps depend value given gaps scaled log compared log visually different values deviation log notable near boundaries convergence also slower particularly behavior consistent divergences approaches explicit interval bounds thm cor comparison confidence intervals let one pch ppbr given value confidence set determined test monotonicity properties closure interval compute endpoint numerically inverting exact expressions example shown fig show endpoints according test function tests endpoints converge number trials grows notably relative separation endpoints large level quantify behavior theqendpoints different tests normalize empirical standard deviation defined empirical endpoint deviation exact test large expect quantity determined tail probabilities standard normal distribution significance probability normal variance exceeds expect take point view performance test characterized size endpoint deviation relative size deviations two tests close perform similarly purpose characterizing parameter another way comparing intervals obtained consider coverage probabilities situation coverage probability test approximated determining thm one infer coverage probability approximately coverage probabilities conservative larger particularly small ppbr determined interval bounds empirical endpoint deviation three tests details app vii next theorem summarizes results asymptotically theorem let log negative logarithm tail standard normal fix write log constant depending satisfies pch ppbr log log last expression following approximation relevant sufficiently large log log log log relative error approximation first two identities goes zero grows case last identity relative error large dominated term log large required small relative error proof expression pch obtained thm appendix changing relative approximation errors absolute errors obtain expression ppbr note term thm satisfies log log see thm square root pulls max log term dominated assumption obtain expressions refer thm lower bound implies log intervals thm give relative errors need converted absolute quantities positivity monotonicity sufficiently large positive constants explicit values obtained thm simplified argument absorbing additive terms theorem term constant chosen sufficiently large consider sufficiently large expression denominator approximation error side exceeds constant multiple new constant order notation simplifies suffices apply see proof obtained bound via bounds expression log log log log log statement thm bounds two inequalities log log log log log log log let log log monotonically increasing first inequality implies need bound form conclude bound type obtained appendix definiteness restrict show bound holds suffices establish since log log log log finish argument apply inequality given bound becomes get equivalently applying monotone sides gives let satisfy write log log write fixed point equation log accomplish goal determining lower upper bounds fixed point since iteration alternating around fixed point provided provided conclude since assuming condition satisfied implies region iteration converges bounds require bound according follows log log log log log log log log log log log log log get log log applying definitions log log approximation error obtained expanding square root could used newton method starting obtain better approximations one step resulting expression involved expression confirms expectation approaches expected value standard normal distribution may compared theorem empirical endpoint deviation test approaches exact test small large squares differ term order log log log notably ratio pbr tests empirical endpoint deviation grows log relationships visualized figs different values figures show theprelative sizes empirical endpoint deviations tend toward smaller log relative growth pbr test endpoint deviations leads less doubling deviations relative exact test even test coverage probabilities much closer nominal value believe lead unreasonably conservative results many applications next consider behavior true endpoint deviations given normalized differencepof true success probability endpoint obtained one tests let true standard deviation define true endpoint deviation determined test true endpoint deviations show inferred endpoint compares therefore directly exhibits statistical fluctuations contrast empirical endpoint deviations lowest order independent take view two tests endpoints perform similarly true endpoint deviations differ amount small compared width distribution true endpoint deviations compare three tests basis consider quantiles corresponding gaussian standard deviations constant quantiles satisfy theorem thm since also see substituting definition implicit constants depend large exact tests endpoints close dominated perform similarly hold comparison exact test endpoints pbr test since latter endpoint deviation grows log pbr test robustness stopping rules requires endpoint deviations must grow qualitatively expect growth least log log due law iterated logarithm growth slower log growth found suggesting improvements possible observed ref many applications number trials acquired determined ahead time full robustness stopping rules necessary however ability adapt changing experimental conditions may still helpful example sect shows know number trials ahead time retain ability adapt avoiding asymptotic growth endpoint deviations pbr test strategy avoiding asymptotic growth pbr test endpoint deviations set aside first trials training infer probability success use determine test factor used remaining trials strategy endpoint deviations bounded average typically formalize training strategy follows modify setting let otherwise valid test factors null testing given defined call test define investigate behavior quantities corresponding rvs function consider values maximized monotone either side max use instead without changing endpoint calculations determining endpoint confidence interval seek maximum maximum value satisfies min follows location maximum implies show endpoint deviations test typically constant factor larger test large factor approaches approximating endpoint deviations test trials begin comparing pch latter applied last trials use place log difference log log pch nkl expanding second order log log log log max let expansion values max rvs independent means variances furthermore asymptotically normal variances consequently asymptotically normal variance accordingly probability asymptotically given tail standard deviations standard normal determining typical behavior consider constant asymptotic purposes observe probability notation subsumes polylogarithmic factor law iterated logarithm write let fix level thereby also log define smallest solution log log log pch pch estimate subscript makes previously implicit number trials explicit finish comparison express endpoint relative write considering case therefore identify compares promised pch conclusion clear specific problem hypothesis testing confidence intervals bernoulli rvs always preferable use exact test ideal case trials general nulls exact tests typically available approximations used approximations often take account failure underlying distributional assumptions approximation errors large high significance thus trustworthy alternatives based large deviation bounds test supermartingales desirable goal suggest alternatives better example bernoulli rvs determine gap exact test case exact test known tests readily calculable suggestion high significance applications gaps relatively small relevant logarithmic scale within expected variation even moderate significances confidence intervals increase size bounded constant number trials known ahead time slowly growing cost number trials allow arbitrary vii appendix preliminaries notation definitions introduced text bounds obtained three tests investigated denoted exact pch ppbr pbr test depend reference definitions otherwise ppbr otherwise pch gain per trial bound log values usually constrained unless otherwise stated assume appendix dedicated obtaining upper lower bounds log lower bounds endpoints confidence intervals make sure upper lower bounds differ quantities converge zero grows differences log confidence lower bounds generally aim simplicity expressing bounds obtain tight constants expression ppbr theorem define proof proof proceeds induction write side side evaluates required usual convention empty product evaluates suppose holds trial use expand binomial expression rewrite side since identity simplifies thus establishing induction step expression seen inverse positive martingale determined complete family martingales obtained ville chapter sect obtained ville probability measure approximations use log pch nkl reference value according thm log ordered according log ppbr log pch log express asymptotic differences log use auxiliary functions first log log log log log first two terms expression recognized shannon entropy independent random bits bit value probability stirling approximation applied binomial coefficient get log log log log log log log log increase interval simplify bounds preserving convergence large lower bound use upper bound note maximized therefore increase upper bound according obtain interval expression log log valid boundary values log next auxiliary function bounds ref see reference summary properties mentioned ref details function related tail standard normal distribution function pby monotonically decreasing convex satisfies differential equation make use following bounds involving log lower bound comes upper bound upper bound lower bound specifically compute log log log log log definitions express log terms difference log pch theorem log log pch log log log log ppbr log pch proof theorem obtained substituting definitions applying bounds details log ppbr log log log log pch log log log pch log remains substitute interval expression theorem define len min log log ppbr log log len log log log pch log log len log observe len respect constant first term defining minimum smaller within less one standard deviation defined primary dependence parameters visible interval bounds proof approximating apply thm ref following sequence substitutions first four expand definitions reference given substitutions defined obtain log log len log log log len log log ppbr log log len log second identity theorem follows substituting expression thm eliminate function expressions applying bounds corollary assumptions thm len log log pch log log proof define view log log log log log substituting simplifying expression gives desired result asymptotic normality log differences main tool establishing asymptotic distribution log delta method version sufficient purposes thm cor ref means converges distribution normal distribution notation mean variance central limit theorem satisfies application delta method therefore yields next lemma lemma let differentiable theorem pch ppbr constant gain per trial converges distribution according log proof consider pch first lem define log log pch derivative get log theorem follows pch applying lem thm law large numbers imply log ppbr log pch converges probability cor implies namely log log pch converges probability general converges probability see ref prop statement theorem proven follows comparison ngn ppbr pbr ngn ngn pch differences log much tighter distributions also asymptotically normal scaling variances given next theorem differences log standard deviations theorem let constant ppbr npch satisfies ppbr log npch log npx satisfies npx pch proof thm law large numbers see ppbr log log npch converges probability zero lem log conclude log log combining observations gives similarly cor taking note definition len npx log log pch converges probability zero relevant derivative log log log combining two observations gives confidence interval endpoints confidence intervals need determine lower boundaries acceptance regions confidence lower bounds monotonicity suffices solve equations form log desired significance level obtain lower upper bounds solutions illuminate asymptotic behavior solutions log reparametrize qthe scale set empirical standard deviation namely thus express solution terms scaled deviation inverting get theorem let suppose solution identity log pch satisfying constants theorem elsewhere chosen convenience optimality better constants extracted proofs note upper bound ensures reciprocal square root bounded away zero however relative error zero grows requires proof consider parametrized bound later set match theorem statement let log pch continuous monotone increasing standard simplification bound noted ref pch solving desired equation monotonicity turn bounded according according qbound conclude solution get assumed taylor expand remainder third order around write restrict according derivatives written explicitly follows since use bounds bound remainder taylor expansion get cleaner expressions decrease denominators substituting gives log pch nkl could taken lower bound interval zero theorem prefer separate cases substitute bound multiplying interval use lower bound interval inequality theorem inverting inequality gives substituting bound multiplying interval gives monotonicity appropriate operations theorem statement simplify bounds theorem let define log suppose solution identity log ppbr satisfying proof thm log pch log ppbr define solving log ppbr equivalent solving log pch since depends depend therefore apply thm get desired bounds theorem let log suppose log solution identity log satisfying max extend negative values necessary evaluating interval expression function negative logarithm tail standard normal distribution lower bound thm ensures solution log constants multiplying reference interval expressions note large limit terms negligible value thm corresponds standard normal monotonicity explicit bounds obtained combining lower upper bounds intervals expression remark behaves well respect relative error large enough inequalities log log establish implicit differentiation properties noted therefore log integrate log show log log making use identity log consider since monotone increasing integral lower bound gives inequality integration monotonicity determine relative error write obtain interval inclusion interval relationship weakened relative error side given term multiplying interval written relative error bounded twice relative error course interval bounds converge need proof proof thm consider parametrized bound later set match thm statement ofq bound get define start rewritten follows log nkl log log log len len better bounds small values use alternative definition len according lower bound last interval len next approximate nkl terms instead still write interval bounds terms let concerned range log log yielding nkl fourth term side log whose absolute value bounded given range thus log since pch also use bound obtained proof thm substituting needed equation solve log log sum first three terms evaluates remaining terms independent order merged means common bounds using since standing assumptions integer different consequently interval bounds combine conservatively write holds iff monotonicity extending negative arguments mentioned statement thm necessary constraint equivalent log know add max theorem statement determine interval equation use remainder bound factor side consider numerator write bound get theorem follows composing constraint acknowledgments work includes contributions national institute standards technology subject copyright would like acknowledge supports ontario research fund orf natural sciences engineering research council canada nserc industry canada masanes certified randomness quantum physics nature bierhorst robust mathematical model experiment phys brockwell davis time series theory methods springer verlag new york edition herman chernoff measure asymptotic efficiency tests hypothesis based sum observations annals mathematical statistics christensen hill kwiat knill nam coakley glancy shalm zhang analysis loopholes experimental bell tests phys rev sep tommaso dorigo extraordinary claims solution epj web conferences volume page edp sciences marco genovese research hidden variable theories review recent progresses physics reports giustina test bell theorem entangled photons phys rev dec hensen bell inequality violation using electron spins separated nature hoeffding probability inequalities sums bounded random variables journal american statistical association kullback leibler information sufficiency ann math larsson loopholes bell inequality tests local realism phys brendan mckay littlewood estimate binomial distribution adv appl nagaev chebotarev bound proximity binomial distribution normal one doklady mathematics jagdish patel campbell read handbook normal distribution volume crc press wenjamin rosenfeld daniel burchardt robert garthoff kai redeker norbert ortegel markus rau harald weinfurter using entangled atoms simultaneously closing detection locality loopholes phys rev glenn shafer alexander shen nikolai vereshchagin vladimir vovk test martingales bayes factors statist shalm strong test local realism phys rev dec jun shao mathematical statistics springer new york edition ville etude critique notion collectif paris yanbao zhang scott glancy emanuel knill asymptotically optimal data analysis rejecting local realism phys rev dec yanbao zhang scott glancy emanuel knill efficient quantification experimental evidence local realism phys rev nov figures log median log pch quantiles log pch median log median log pch median log ppbr median log pch quantiles median log pch fig comparison log top half figure shows median quantile log pch median agrees symmetry lower half shows median differences log log pch ppbr difference quantiles median log pch also shown within range plot even small dominate median differences except approaches absolute longer extremely small log log pch log log ppbr log pch log log log pch log log ppbr log pch log log log pch log log ppbr log pch log log log pch log figures fig gaps log show malized differences log log pch log pch large constant pbr test normalized difference converges exact test converges horizontal lines indicate limit lowest order normalized asymptotic differences log diverge pch ppbr figures fig lower endpoints level confidence set three tests function figures pch ppbr asymptotic quantile unit pch asymptotic ppbr asymptotic fig empirical confidence set endpoint deviations level function continuous lines show expressions obtained dropping terms exact test expressions normal approximation therefore match absolute value quantile unit normal figures pch ppbr asymptotic quantile unit pch asymptotic ppbr asymptotic fig empirical confidence set endpoint deviations level function see caption fig figures pch ppbr asymptotic quantile unit pch asymptotic ppbr asymptotic fig empirical confidence set endpoint deviations level function see caption fig
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imnet imperative network programming language mohamed computer information sciences imam mohammad ibn saud islamic university imsiu riyadh kingdom saudi arabia feb college adel alsalem college computer information sciences imam mohammad ibn saud islamic university imsiu riyadh kingdom saudi arabia alsalem department mathematics faculty science cairo university giza egypt maelzawawy abstract one recent architectures networks networks sdns using controller appliance control set switches network controlling process includes installing uninstalling rules flow tables switches paper presents imperative network programming language called imnet facilitate writing efficient yet simple programs executed controller manage switches imnet expressive compositional imperative paper also introduces operational semantics imnet detailed examples programs operational semantics constructed imnet illustrated paper well network programming languages architecture operational semantics syntax imnet introduction network group appliances connected exchange data among appliances switches forwarding data depending mac addresses routers forwarding data depending addresses firewalls taking care forbidden data network appliances connected using model efficiently allows forwarding storing ignoring tagging providing statistics data moving network network appliances like routers special functionality control network enables routers compute determine routes data network course different networks different characteristics abilities open networking foundation suggested removing control owned different network appliances adding instead generalpurpose appliance controller program different network appliances querying data flowing network impact simple suggestion huge giant networks need corresponding author complex expensive switches networks cheap programmable switches used programmed configure optimize networks via writing programs running controllers networks sdns networks established using architecture precise implementation architecture openflow used achieve various applications monitoring data flow balancing switch load network management controlling appliances access detection service absence host mobility forwarding data center therefore sdns caused appearance network programming languages paper presents imnet imperative highlevel network programming language imnet expresses commands enabling controllers program network appliances including switches imnet clear syntax based classical concepts imperative programming allows building rich robust network application natural way imnet realized generalization frenetic functional network programming language clear fact core programs written imnet frenetic based query result form stream values packets switches ids commands treating packets imnet include constructing installing adding flow tables switches switch rules imnet supports building simple programs express complex dynamic functionalities like load balancing authentication imnet programs also analyze packets historical traffic patterns motivation motivation paper lack simple syntax imperative network programming language yet stronger motivation existing network programming languages supported theoretically using operational semantics type systems program logics like floydhoare logic contributions contributions paper following definition types int switch ids packet switch ids int bool types values natural numbers switch ids packets switch ids natural numbers boolean values values expression denotes type value events sendcontroller change sendall sendout rules patterns actions rules switch ids rules switch ids new syntax imperative network programming language imnet var operational semantics form states inference rules constructs imnet states two detailed examples programs constructed imnet precise operation semantics program imnet sequence queries followed statement result query event finite sequence values event concept also used frenetic however event frenetic infinite sequence values value integer switch packet triple switch integer boolean value pair two values value type set types paper focus details statements interesting part network programming language possible actions taken certain switch certain packet sendcontroller sendall sendout change action sendcontroller sends packet controller take care action sendall sends packet switches action sendout sends packet switch certain port action change modifies header field packet new value rule semantics pair pattern action pattern form concretely describes set packets action action taken elements set packets rules stored tables called flow tables switches represents initial assignment rules flow tables switches organization rest paper organized following section presents syntax semantics imnet proposed semantics operational hence consists states inference rules presented section two detailed examples programmes built imnet presented section section also explains two examples assigned precise semantics using proposed operational semantics section reviews related work gives directions future work section concludes paper semantics section presents syntax semantics imnet programming language sdn networks using architecture figure shows syntax imnet figures present semantics imnet constructs proposed semantics operational states defined following definition lvar eventrans queries lift applylft applyrit merge mixfst mixsnd filter makforwrule makerule stmts addrules register send defs progs figure imnet syntax state proposed operational semantics triple triple captures current state program variables hence map set variables set events rule lists imnet variables may contain events rule lists symbol captures current state flow tables switches hence map switches ids rule lists finally initial assignment rules assigned switches registered yet added yet five type statements imnet assignment statement assigns result event transformer variable statement addrules adds switch rules stored reservoir initially assigned rules rules assigned switches added flow tables yet statement register makes initial assignments permeant adding flow tables switches statement send sends specific packets treated certain way certain switches keep record actions takes packets different switches assume map called history set switches ids set lists pairs packets taken actions map used rule sends operational semantics statements given figure judgement inference rules figure form judgement reads following execution state ends execution reaches state inference rules figure use figure get semantics important construct imnet event transformers judgements figure form meaning semantics transformer variable state event transformer lift applies map values event rule lifts event transformer filter filters event using map rule filters given set actions two events event transformers mixfst mixsnd create lists rules rules controller programs section presents two examples programs constructed using syntax imnet figure first example constructs rules based information stored variable installs established rules flow tables switches stored program following statements makerule lift addrules register first statement program makes rule value event stored second statement assigns rules switch ids event stored third statement stores rule assignment initial rule assignment last statement program adds established rules flow tables switches figure shows operational semantics program using semantics previous section second example constructs forwarding rules based source ips arriving packets installs established rules flow tables switch ids stored program following statements sourceips applylft port lift switch makforwrule addrules register lifts lift merges merge true filters filter applylft applyrit type types onces times mixfst mixsnd makforwrule sendout sendout sendout makerule vni mfrs mkrls figure operational semantics event functions imnet first statement program assumes function sourceips returns source ips arriving packets stores form event second statement transfers event event pairs ips port numbers packets forwarded third statement augments values event switch ids event stored fourth statement makes forward rule value event stored fifth statement stores rule assignment initial rule assignment last statement program adds established rules flow tables switches figure shows operational semantics program using semantics previous section related future work section presents work related presented current paper one early attempts develop defined networking sdn nox based ideas nox uses explicit callbacks rules examples applications benefitted nox load balancer work many directions improving platforms programming networks include maestro onix uses distribution parallelization provide better performance scalability famous programming language networks frenetic two main components first component collection operators operators aim establishing treating streams network traffic operators also built concepts functional programming query languages declarative database moreover operators support modular design cost control semantics singletier programming declarative design second component frenetic system system facilitates actions adding assgns seqs addrls addrules regs register history sends send figure operational semantics statements imnet moving rules flow tables switches one advantage imnet language presented paper frenetic imnet imperative therefore imnet paves way appearance types network programming languages network programming langues network programming languages examples program network components though languages ndlog netcore netcore provides integrated view whole network ndlog designed explicitly distributed fashion extension datalog ndlog presented determine code protocols routing overlay networks concepts like hash tables distributed systems imnet presented paper frenetic ndlog classified highlevel network programming languages ndlog main focus overlay networks routing protocols frenetic functional way imnet imperative way focus implementing packet processing modifying header fields therefore imnet equips network programmer modular view network provided ndlog frenetic supported fact program ndlog single query calculated router network switch component networks programmed via many interfaces openflow platform examples platforms include routebricks click modular router snortran bro idea produce certain hardware programs achieves routebricks stock machines used improve performance program switches modular approach platform click modular router enables programming network components system focuses software switches form linux kernel code sake intrusions detection preserving network security snortran bro enable coding monitoring strategies robust one advantage imnet language presented paper related work imnet overcomes disadvantage similar languages focusing controlling single device many interning directions future work one direction develop methods static analysis network programming languages obviously associating analyses correctness proofs spirit many network applications conclusion networks sdns recent architectures networks controller device programs network devices specially switches via sequence installing uninstalling rules memories devices paper presented imperative network programming language called imnet facilitate job controller efficient yet simple programs imnet advantages simplicity expressivity propositionally imperative paper also introduced concrete operational semantics meanings imnet constructs detailed examples using imnet operational semantics also illustrated paper srcport sendall inport sendcontroller makerule srcport sendall inport sendcontroller srcport sendall inport sendcontroller lift srcport sendall inport sendcontroller srcport sendall inport sendcontroller addrules srcport sendall inport sendcontroller srcport sendall inport sendcontroller register srcport sendall inport sendcontroller srcport sendall inport sendcontroller figure example operational semantics program written imnet sourceips applylft port lift switch makforwrule sendout sendout addrules sendout sendout sendout sendout register sendout sendout sendout sendout figure example operational semantics program written imnet references loo hellerstein stoica ramakrishnan declarative routing extensible routing declarative queries sigcomm cai cox maestro system scalable openflow control technical report rice university dec koponen casado gude stribling poutievski zhu ramanathan iwata inoue hama shenker onix distributed control platform production networks osdi oct handigol seetharaman flajslik mckeown johari loadbalancing web traffic using openflow demo acm sigcomm aug heller seetharaman mahadevan yiakoumis sharma banerjee mckeown elastictree saving energy data center networks nsdi apr wang butnariu rexford server load balancing gone wild mar greenberg hjalmtysson maltz myers rexford xie yan zhan zhang clean slate approach network control management sigcomm ccr october casado freedman pettit luo gude mckeown shenker rethinking enterprise network control trans networking aug gude koponen pettit pfaff casado mckeown shenker nox towards operating system networks sigcomm ccr foster guha reitblatt story freedman katta monsanto reich rexford schlesinger walker harrison languages networks ieee communications magazine bain campbell karlsson modeling growth dynamics neural networks via message passing erlang neural models natural home message passing functional programming languages erlang workshop flow pointer analysis based memory safety multithreaded programs murgante gervasi iglesias taniar apduhan eds iccsa part lncs vol springer heidelberg probabilistic pointer analysis multithreaded programs scienceasia detection probabilistic dangling references programs using tools iccsa frequent statement dereference elimination distributed programs iccsa suzuki pinte cutsem meuter yonezawa programming language support routing pervasive networks percom workshops elsts selavo approach wireless sensor network programming languages sesena monsanto foster harrison walker compiler system network programming languages popl hong joung meso programming language building stronglytyped network applications sac rexford programming languages programmable networks popl arneson langbort linear programming approach routing control networks constrained linear positive systems automatica loo condie hellerstein maniatis roscoe stoica implementing declarative overlays sigops loo hellerstein stoica ramakrishnan declarative routing extensible routing declarative queries sigcomm foster harrison meola freedman rexford walker frenetic langauge openflow networks presto nov foster harrison freedman monsanto rexford story walker frenetic network programming acm sigplan international conference functional kohler morris chen jannotti kaashoek click modular router acm transactions computer systems aug egorov savchuk snortran optimizing compiler snort rules fidelis security systems paxson bro system detecting network intruders realtime computer networks dec dobrescu egi argyraki chun fall iannaccone knies manesh ratnasamy routebricks exploiting parallelism scale software routers sosp oct mckeown anderson balakrishnan parulkar peterson rexford shenker turner openflow enabling innovation campus networks sigcomm ccr chen lian lin liu liu achieving high performance compiled network applications enabling ease programming pldi jun cristea zissulescu deprettere bos towards language support reconfigurable packet processing samos jul open networking foundation mar see http
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codon context optimization synthetic gene design dimitris papamichail hongmei liu vitor machado nathan gould robert coleman georgios papamichail novo synthesis dna computational gene design methods make possible customization genes direct manipulation features codon bias mrna secondary structure codon context another feature significantly affecting mrna translational efficiency existing methods tools evaluating designing novel optimized protein coding sequences utilize untested heuristics provide quantifiable guarantees design quality study examine statistical properties codon context measures effort better understand phenomenon analyze computational complexity codon context optimization design exact efficient heuristic gene recoding algorithms reasonable constraint models also present tool evaluating codon context bias appropriate context index computational biology dynamic programming simulated annealing synthetic biology introduction xpression genes crucial cell activity major determinant phenotype derived genotype inherently fundamental modern biotechnology expression process information gene used synthesis functional gene product often protein several steps gene expression process may modulated including transcription rna splicing translation posttranslational modification protein concentrate process translation effect synonymous mutations gene confer expression corresponding protein method traditionally employed modify posttranscriptional expression genes working towards objectives synthetic biology precise protein expression control direct implications improving heterologous expression successfully designing finetuning gene regulatory networks recent years seen significant advances toward understanding control rate translation genes newfound knowledge led development number algorithms computational tools aim enable life scientists create synthetic genes constructs first generation design tools focused primarily optimizing designs papamichail department computer science college new jersey pennington road ewing usa email papamicd liu division biostatistics department public health science miller school medicine university miami miami usa machado gould department computer science college new jersey pennington road ewing usa coleman department biology farmingdale state college suny farmingdale usa papamichail department informatics new york college amalias athens greece manufacturability oligos without local secondary structures end repeats instead biological activity soon oligo design process separated gene optimization process new tools emerged address two processes separately codon bias characteristic pattern preference usage synonymous codons highly lowly expressed genes shown correlated expression levels gene dominant method designing synthetic genes translational efficiency codon context bias emerged another critical feature affecting gene translation rates independently codon bias less well studied understood latter current methods tools used optimize codon context gene rely heuristics provide guarantees optimality study examine statistical properties codon context bias effort better understand phenomenon measures used quantify also describe analyze exact algorithms designed creating synthetic genes optimized codon context evaluate optimization potential simulated annealing heuristic method choice toward efficient gene customization finally make available webbased utility enable characterization gene codon pair bias providing statistical information optimality background codon bias species synonymous codons used unequal frequencies codon usage bias recognized crucial shaping gene expression cellular function affecting diverse processes rna processing protein translation protein folding rarely used codons associated rare trnas shown inhibit protein translation favorable codons opposite effect something particularly pronounced prokaryotic organisms process substituting rare codons favorable ones referred codon optimization controlling codon bias without considering optimization objectives order modulate translation rates computationally easy since involves certain synonymous substitutions reach desired distribution quantification effect though much difficult experimentally use particular codons synonymous mutations shown certain cases increase expression transgenes genes expressed heterologous host several different statistical methods proposed used analyze codon usage bias methods frequency optimal codons fop codon adaptation index cai trna adaptation index tai used quantify codon preferences towards codons predict gene expression levels alternative methods effective number codons enc shannon entropy information theory used measure codon usage evenness relative synonymous codon usage rscu synonymous codon usage order scuo additional measures latter category optimization codon bias singular objective algorithmically straightforward performed asymptotic linear time space function sequence length worst average cases true maximization minimization towards given codon bias measure cai rscu enc etc well given codon distribution including case codon position assignments performed randomly codon context bias gutman hatfield first noticed codon pairs prokaryotic genes exhibit another significant bias towards specific combinations studies revealed codon pair optimization influences translational elongation step times functional significance studied small datasets recent work synthesized novel coding regions utilizing large scale codon pair optimization coupled novo synthesis constructs experimentation provided evidence influence codon pair bias translational efficiency several mathematical methods proposed study codon context bias including aware least three existing gene design tools date provide functionality controlling codon context computational complexity optimization codon context singular objective asymptotic linear time complexity worst case function sequence length shown subsection optimization codon pair bias fixed codon distribution considerably harder although polynomially time solvable section latter problem reduced ation travelling salesperson problem solved dynamic programming algorithm time space asymptotic complexity length sequence optimized consequence currently available tools attempt codon context optimize synthetic genes often conjunction objectives utilize metaheuristics simulated annealing genetic algorithms heuristics guarantee optimal solution limit running time optimization procedure typically computing reasonable approximations codon context optimization software several published gene design tools provide functionality controlling codon context albeit two tools share measure codon context bias eugene standalone tool developed gene optimization program provides graphical user interface connects databases genbank retrieve genomic sequences based sequence identifiers among numerous features eugene provides functionality optimize mrna codon context bias codon autocorrelation eugene uses indicate improvement towards target objective instead scores makes difficult one interpret compare results software provides two optimization methods simulated annealing genetic algorithm former significantly efficient codon optimization online cool utility optimize multiple objectives including codon context bias optimization process uses genetic algorithm produce several approximately paretooptimal solutions given set design criteria randomly generated sequences evaluated ranked mutated stability threshold reached point fittest sequences based chosen properties outputted algorithm terminates codon context optimization based matching given host codon pair distribution cumulative score available quantify end result current methods tools severe limitations crucial lack reference information optimization objectives optimality designs arbitrary scores used quantify codon codon context bias hard justify use one method another following sections focus codon context study statistical properties well exact approximate methods evaluate quality optimized design allow put codon context right context statistical properties codon context bias phenomenon unequal usage synonymous codons observed genes called codon bias widely used measure quantifies far codon usage gene departs equal usage synonymous codons effective number measure synonymous codons used gene codon usage bias independent amino acid composition gene length calculated estimated homozygosity estimated homozygosity distribution codon pair bias first show pendent different codon pairs frequency usage synonymous codons obtained dividing actual usage total usage amino acid values amino acids added together get effective number codons vary whole gene thus value one codon used amino acid codons used equally amino acid codon pair bias cpb refers phenomenon synonymous codon pairs used less frequently expected example amino acid pair alaglu expected encoded gccgaa gcagag equally frequently based codon frequencies fact gccgaa used frequently gcagag human genes supporting material measure cpb defined following formula codon pair score cps follows codon pair encodes amino acid pair denotes frequency observed number occurrences expected number occurrences cps given codon pair indicates whether pair given genome codon pair bias cpb entire gene sequence arithmetic mean codon pair scores pairs making entire gene sequence analysis follows estimate distribution cpb human genes based calculate pvalue protein encoded mrna specific cpb value also demonstrate codon pair bias independent codon bias amino acid bias length gene sequence measured codons definitions cps scores possible codon pairs human genes excluding stop codons calculated using consensus cds ncbi reference sequence database release date march supplementary material cpb values main set consistently annotated human genes also calculated figure indicates prevalent use codon pairs human genes mean codon pair bias annotated human genes also peak distribution figure examined statistical properties codon pair scores population mean variance consistently annotated human protein coding gene set log similarly log since independent approximately independent independent mutually independent distribution though unknown central limit theorem approximately normally distributed large mean cps variance var cps length gene sequence measured codons definition population mean variance cps weighted arithmetic mean possible codon pairs excluding stop codon human orfeome cps var cps calculated var based data found supplementary material cps human genes equal var cps thus normally distributed mean variance results match figure cpbs calculated core set consistently annotated human genes using formulas cps cpb protein encoded mrna specific codon pair bias calculated two tail cumulative probability exp let probability threshold significance selected log base variance cps larger vary larger range reducing effect small numerical rounding errors resulting values discounting codon bias expected occurrences given codon pair calculated assumption codons independent independent ratio much larger smaller indicates nonrandom utilization codon pairs since nonrandom usage codons related independence effect codon pair scores thus contribute codon pair bias definition context dnorm thus normalized cps codon pair figure cpb distribution genes length based human orfeome implies probability length protein encoded mrna codon pair bias rare significant interval reference examples significant interval significant interval significant interval independence codon context bias amino acid codon bias discounting amino acid pair bias randomly occurring amino acid pairs protein sum expected numbers codon pairs encoding amino acid pair amino acid pairs total equal sum observed numbers gutman hatfield codon pair encoding amino acid pair group codon pairs encoding amino acid pair expected values normalized multiplied normalizing coefficient way sum expected values observed values group algorithms codon context optimization section examine problem designing mrna encoding given protein optimizes codon context bias optimization occur either towards maximization minimization given measure quantifies codon context bias codon pair bias defined coleman codon context bias defined moura normalized offset value defined boycheva codon context bias measure bounds span nucleotides context codon constant subsequent discussion consider problem bias maximization minimization achieved exact manner concentrate codon pair score cps codon pair bias cpb defined equations respectively measures choice quantify codon context bias supporting experimental evidence validity familiarity computation codon pair bias optimization without codon restrictions designed implemented algorithm compute maximum minimum cpb proteincoding gene polynomial time using dynamic programming given protein coding gene length whose amino acid composition calculate mrna sequence maximum similar minimum cpb encoding protein follows starting amino acid process amino acid order keeping track highest cpb mrna prefix ending codon number synonymous codons encoding amino acid let cpbi highest score mrna prefix ending codon position addition codon keep track ent codon encoding amino acid led figure dynamic programming algorithm codon context optimization highest cpb prefix score ending codon step algorithm proceeding amino acid form possible codon pairs encoding amino acid pair compute maximum cpb mrna prefixes ending codon max cpbi cps store value codon together appropriate parent codon notation cps used instead cpsab clarity amino acids processed select codon highest cpb score among synonymous codons value highest cpb mrna encoding gene mrna encoding highest cpb generated following parent pointers toward start amino acid chain concatenating appropriate codons outline data structure used observed figure theorem computes optimal cpb protein coding gene worst case time space complexity length amino acid sequence gene proof algorithm correctness sufficient prove proposition identifies mrna highest cpb translating given protein integer proposition follows strengthened proposition easier prove using induction identifies mrna highest cpb translating given protein ending codon number synonymous codons translating amino acid base case first step algorithm codon pair highest cps encoding ending codon formed following parent pointer concatenate two codons codon pair formed highest cpb among synonymous ones since highest cpb value stored codon position therefore true inductive step assuming proposition true arbitrary fixed integer show true claim step compute store codon highest cpb score mrna encoding prefix protein ending position last codon true since possible codon pairs considered concatenated optimal mrna encodings ending codons according inductive hypothesis selecting mrna sequence ending codon highest cpb score among storing score appropriate parent pointer codon allows identification mrna sequence optimal cpb ending codon therefore true integers follows also true linear time space complexities follow linear number steps algorithm takes constant number codon pairs processed since maximum number synonymous codons amino acid constant amount information stored namely cpb values prefix ending given synonymous codon parent pointers quantities bounded constant polynomial cpb optimization fixed codon distribution due significant effect codon bias mrna translation rates important control codon bias recoding amino acid sequence optimizing codon pair bias reason concentrate optimizing cpb assigning fixed codon distribution mrna sequence remains unchanged cpb optimization procedure problem maximizing presence codon pairs mrna sequence keeping amino acid sequence codon frequency distributions intact reduced variant tsp travelling salesperson problem particular variant thought computing shortest route travel series countries amino acids prespecified sequence selecting city codon visit country given fixed distribution cities polynomially solvable using dynamic programming time complexity number codons examined mrna sequence describe another algorithm polynomially solves aforementioned problem algorithm similar cpb optimization without codon restrictions particular case though keep track optimal cpb mrna prefixes ending synonymous codons amino acid position also allocated codons prefixes future codon choices may limited based selections made far use dynamic programming reduce exponential number codon permutations utilizing fact knowledge specific codon assignments position necessary total number codons type already allocated restrict selections sufficient keep track codon frequencies optimal cpb every prefix ending specific codon used specific codon distribution since codon may placed mrna protein maximum times codons amino acid position keep track maximum values algorithm executes total steps resulting time complexity space utilization time complexity algorithm reduced considering synonymous codon degrees freedom every amino acid encoded synonymous codons need keep track frequencies codons since frequency last codon derived rest improved time bound therefore result degrees freedom codons similarly space required algorithm even algorithm despite polynomial practical except smallest protein coding genes correctness algorithm argued similar manner proof theorem branch bound cpb optimization fixed codon distribution study effectiveness heuristics optimizing cpb synthetic genes fixed codon distributions need reference algorithm produces optimal encodings compare approximate solutions designed branch bound algorithm calculate maximum minimum cpb given gene given codon distribution examines possible mrna encodings gene fixed codon frequencies algorithm simply enumerates possible encodings assigning amino acid order corresponding codon pool available codons performs depth first search space codon assignments position amino acid chain explored tree height internal node maximum children process depicted figure eliminate branches leading provably solutions compare best gene cpb result thus far potential current prefix cpb added optimal cpb suffix mrna determined considers possible codon assignments cpb maximization initial bound arbitrarily negative number previously computed design method simulated annealing described section optimal cpb suffixes mrna running dynamic programming starting end amino acid chain storing intermediate results array provides constant lookup time optimal cpb codon distribution suffix given gene asymptotic time complexity exponential function protein length remains figure branch bound cpb optimization algorithm fixed codon distribution tial practice demonstrated section worstcase space complexity linear owing depth first search process constant amount information stored step algorithm addition precomputed suffix cpb upper bound scores simulated annealing cpb optimization fixed codon distribution simulated annealing metaheuristic well suited problems particularly useful problems synthetic gene design since effectively applied optimize mrna sequence multiple objectives addition codon pair bias restriction site rna folding energy manipulation used simulated annealing extensively past design synthetic viral bacterial genes featured experimented conflicting simulated annealing optimizing polio virus capsid protein encodings codon pair utilization cpg content unpublished data codon pair bias effect translation determined independent cpg content coding sequences least rna encoding capsid protein polio virus simulated annealing procedure works follows initially create random assignment codons respective amino acid allowed positions new codon distribution defined first assign codons amino acid according new distribution create random assignment calculate codon pair bias coding sequence initial assignment randomly select amino acid randomly consider two locations amino acid encoded coding sequence codons positions different calculate sum scores four newly formed codon pairs codons exchanged compare sum original four pairs improvement overall score exchange two codons calculate simulated annealing optimization function decide whether exchange two codons based value way even exchanges probability performed probability varying throughout execution based simulated annealing parameters algorithm progress repeat previous step reached maximum number allowed steps specified execution algorithm experimental results one goals study determine whether simulated annealing heuristic produces cpb designs sufficiently close optimal reduce need run computationally expensive algorithms optimization toward goal implemented two algorithms maximize cpb protein coding genes compare results former producing maximal cpb design latter approximation also implemented aid branch bound algorithm generating necessary upper bounds mrna suffixes cpb scores run algorithmic experiments laptop equipped haswell processor running memory ssd hard drive algorithms run single processor core produced maximal cpb designs prefixes aequorea victoria green fluorescent protein gfp genbank accession number ranged size amino acids upper size limit set computational time constraints grows exponentially function amino acid sequence length utilizes simulated annealing run iterations computational experiment independent protein size algorithm run times input protein prefix best result collected although results input length always optimal running time dependent primarily number iterations experiments presented herein ranged seconds cpb outcomes two algorithms actual running time shown table table cpb optimization algorithm results sequence length aas running time seconds maximal cpb algorithms simulated annealing procedure whose parameters customized experimentally past produce optimized designs poliovirus capsid protein numerous genes variety organisms manages match optimal score gfp prefixes tested amino acids long although expected differences emerge protein size increases reassuring despite already sizable solution space explored tested cases simulated annealing identifies optimal solution belief simulated annealing great choice cpb optimization corroborated cpb scores optimized mrna encodings larger proteins consistently placed large number standard deviations respective mean figure codon context tool cctool input screen codon context evaluation tool created tool compute cpb given gene graphically place score range values determined minimum maximum cpb gene approximated simulated annealing also randomized codon assignments corresponding amino acids generated alternate designs coding protein exact codon distribution approximate mean variance cbp gene example input output screen tool shown figures respectively tool still development accessed online http conclusion codon context bias optimization significantly affects mrna translation rates currently employed several tools design evaluate synthetic protein coding genes due computational complexity figure cctool output page optimizing codon context current methods utilize heuristics provide little information solution optimality addition two methods employ measure codon context results comparable performed statistical analysis better understand codon pair bias experimentally validated codon context measure determined independence amino acid codon biases also designed exact algorithms optimize mrna sequences codon pair bias different constraints run experiments determine performance simulated annealing codon context optimization finally developed webbased utility enable synthetic gene designers evaluate constructs put codon context right context acknowledgment work supported nsf grant muse program college new jersey papamichail corresponding author reached papamicd references welch villalobos gustafsson minshull genes successful protein expression methods vol czar anderson bader peccoud synthesis demystified trends biotechnol vol plotkin robins levine codon usage expression human genes proc natl acad sci vol kudla murray tollervey plotkin determinants gene expression escherichia coli science vol salis mirsky voigt design synthetic ribosome binding sites control protein nat vol welch govindarajan ness villalobos gurney minshull gustafsson parameters control synthetic gene expression escherichia coli plos one vol gould hendy papamichail tools algorithms designing customized synthetic genes frontiers bioengineering biotechnology vol coleman papamichail skiena futcher wimmer mueller attenuation changes codon pair bias science vol lithwick margalit sequencedependent features associated prokaryotic translation genome res vol gustafsson govindarajan minshull bias heterologous protein expression trends biotechnol vol ikemura abundance escherichia coli transfer rnas occurrence respective codons protein genes proposal synonymous codon choice optimal coli translational system mol vol sharp codon adaptation index measure directional synonymous codon usage bias potential applications nucleic acids res vol dos reis savva wernisch riddle codon usage preferences test translational selection nucleic acids vol suzuki saito tomita sum relative entropy new index synonymous codon usage bias gene vol sharp tuohy mosurski usage yeast cluster analysis clearly differentiates highly lowly expressed nucleic acids vol jul wan kleinhofs zhou relationship synonymous codon usage bias composition across unicellular genomes bmc evol vol gutman hatfield utilization codon pairs escherichia coli proc natl acad sci vol irwin heck hatfield pair utilization biases influence translational elongation step times biol chem vol mueller coleman papamichail ward nimnual futcher skiena wimmer attenuated influenza virus vaccines rational design nat biotechnol vol coleman papamichail yano mdel pirofski reduction streptococcus pneumoniae pathogenicity via synthetic changes virulence factor bias infect dis vol fedorov saxonov gilbert codon bias eukaryotic nucleic acids vol boycheva chkodrov ivanov pairs genome escherichia coli bioinformatics vol shah giddings parvaz gesteland atkins ivanov identification putative programmed translational frameshift bioinformatics vol hooper berg genes atypical nucleotide sequence microbial mol vol moura pinheiro silva miranda afreixo dias freitas oliveira santos context analysis codon pairs orfeome genome vol gaspar oliveira frommlet santos moura maximizing synthetic gene design heterologous expression bioinformatics vol chin chung lee optimization cool optimization platform synthetic gene bioinformatics wright apos effective number codons apos used gene gene vol kimura crow number alleles maintained finite population genet vol apr marr wang kam sharma lam markowski solimano lavoie turvey respiratory syncytial type ifn responses human neonates young vol gao wang huang yang guo liu yang feng attenuated deoptimization codon pair bias rna polymerase gene virology vol kirkpatrick gelatt vecchi simulated science vol bertsimas tsitsiklis annealing statistical science vol johnson optimization traveling salesman problem automata languages programming vol paterson springer berlin heidelberg
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heuristic method generate better initial population evolutionary methods khaji mohammadi graduate students complex adaptive systems group school applied physics university gothenburg sweden abstract initial population plays important role heuristic algorithms help decrease time algorithms need achieve acceptable result furthermore may influence quality final answer given evolutionary algorithms paper shall introduce heuristic method generate target based initial population possess two mentioned characteristics efficiency proposed method shown presenting results tests benchmarks introduction evolutionary algorithms eas introduced solve nonlinear complex optimization problems commonly used eas genetic algorithms particle swarms optimization differential evolution method characteristics strengths weaknesses long computational time common drawback schemes specially solution space hard explore hamin maghala many efforts already done accelerate convergence methods works focused introducing improving crossover mutation operators selection mechanisms adaptive controlling parameter settings hamin maghala although population initialization affect convergence speed also quality final solution little reported research field maaranen introduced population initialization genetic algorithms presented results showed proposed initialization method improve quality final solutions noteworthy improvement convergence speed hand generation sequences difficult advantage vanishes higher dimensional problems theoretically dimensions larger also rahmanian hmian maghala presented novel scheme population initialization applying learning make eas faster paper presents heuristic method based combination classical evolutionary methods suggested solving simplified equations complex function addition random individuals make novel initial population converges acceptable answer faster organization rest paper follows section concept learning briefly explained classical briefly reviewed section proposed algorithm presented section experimental results given section results analysed section finally work concluded section benchmark functions listed appendix proposed method let local minimum function every point region dimensional sphere radius center called cavity every function secreted union cavities may empty cavity involves one one local minimum order appropriate initial population generating least one point cavity clearly fruitful although guarantee generate initial population possess characteristic method aims introduce heuristic approach generate initial population cover many cavities main proposed method suppose continuous function initial population approaching function solution following equation theoretically intersections dimensional functions task find point dimensions task seems challenging application since solving called equation possible time suggesting several tricks tackle mentioned difficulty function separation function separated follows calculated points following equations may quality solutions dimension separation function separated follows calculated points following equations may quality solutions experimental results order compare convergence speed suggested method random population initialization der test set numerical benchmark functions employed selected functions global optimization literature test set includes unimodal well highly multimodal minimization problems dimensionality problems varies cover wide range problem complexity range search space also global minimum function given appendix compare convergence speed method der measuring number function calls nfc commonly used metric literature smaller nfc means higher convergence speed theoretical information convergence properties evolutionary algorithms reader referred solved problems lowest positive value one chose also solved problems size initial population sphere model since independently influences hence nfc axis parallel nfc suggested method nfc nfc suggested method obvious less value accuracy result experiments means final answer among initial population rosenbrock function according main function nfc rastrigin function nfc suggested method cosine function hence nfc brainins function nfc suggested method nfc michalewicz function nfc suggested method function nfc nfc suggested method figure performance complete random population population selected initial population michalewicz function runs sort initial population nfc matyas function nfc suggested method inserting nfc nfc suggested method conclusion heuristic method applied famous benchmarks main idea method combine classical arbitrary nature existed methods order generate proper initial population heuristic methods results show many continuous functions final answer obtained selected population suggested method cases population work efficiently within time indicates obtained answers combination random numbers initial population led best result faster case initial population includes random numbers selected answers hence concluded experiments size initial population population including selected results solved equations random numbers outperform include random numbers references evolutionary algorithms theory practice evolution strategies evolutionary programming genetic algorithms university press usa schwefel computational intelligence theory practice new york hammel schwefel evolutionary computation comments history current state ieee transactions evolutionary computation goldberg genetic algorithms search optimization machine learning new york storn price differential evolution simple efficient heuristic global optimization continuous spaces journal global optimization price storn lampinen differential evolution practical approach global optimization natural computing series first maaranen miettinen initial population genetic algorithms computers mathematics applications vol bratley fox niederreiter implementation tests sequences acm transaction modeling computer simulation tizhoosh learning new scheme machine intelligence int conf computational intelligence modelling control andautomation cimca vienna austria onwubolu babu new optimization techniques engineering springer berlin new york rudolph convergence properties evolutionary algorithms verlag hamburg hrstka improvement real coded genetic algorithm based differential operators preventing premature convergence advance engineering software andre siarry dognon improvement standard genetic algorithm fighting premature convergence continuous optimization advance engineering software vesterstroem thomsen comparative study differential evolution particle swarm optimization evolutionary algorithms numerical benchmark problems proceedings congress evolutionary computation ieee publications vol shahryar rahnamayan hamid tizhoosh magdy salama novel population initialization method accelerating evolutionary algorithms computers mathematics applications volume issue may pages
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numerical semigroups nov tizziotti villanueva abstract given positive integer investigate class numerical semigroups verifying property every two subsequent non gaps smaller conductor spaced least semigroups called generalize concept sparse numerical semigroups introduction let set integers numbers set integers subset numerical semigroup closed respect addition complement finite numerical semigroup called trivial numerical semigroup cardinality set called genus numerical semigroup note elements complement called gaps set gaps denoted gaps smallest integer called conductor least positive integer called multiplicity finite set maximum called frobenius number denoted property known number see particular frobenius number consequence fact use notation numerical semigroup denote gaps clear simplicity notation shall write integers danger confusion let numerical semigroup genus gaps said sparse numerical semigroup integer equivalently integer concept sparse numerical semigroup introduced convenience consider numerical semigroup sparse integer let keywords numerical semigroup arf numerical semigroup frobenius variety sparse numerical semigroup tizziotti villanueva case notation clear numerical semigroup genus semigroup called ordinary numerical semigroup canonical examples sparse numerical semigroups work introduce investigate class numerical semigroups verifying property every two subsequent non gaps smaller conductor spaced least positive integer semigroups called generalize concept sparse numerical semigroups paper organized follows arf numerical semigroups introduced since several equivalent properties numerical semigroups appeared section present one properties given prove equivalent characterization given section contains concept leaps numerical semigroup serves basis introducing concept numerical semigroup presented section finally section present results structures numerical semigroups arf numerical semigroups definition semigroup called arf numerical semigroup integers equivalent integers examples arf numerical semigroups ordinary numerical semigroups relevant examples sparse numerical semigroups arf numerical semigroups details arf semigroups see arf property equivalent integers fifteen alternative characterizations arf numerical semigroups distinct given definition goal section prove property arf given equation equivalent characterization given item theorem describe proof recall basics concepts details found two sets contained let let semigroup numerical relative ideal nonempty subset relative ideal contained simply called ideal proper ideal ideal distinct ideal contain zero principal ideal mean stable ideal integer numerical semigroups relative ideal case called relative ideal principal generated relative ideal also relative ideal every relative ideal numerical semigroup ideal called stable principal ideal let numerical semigroup let clear ideal proper ideal theorem let numerical semigroup genus conductor following statements equivalent arf numerical semigroup stable ideal stable ideal integer proof prove initially suppose arf numerical semigroup let positive integer must since clear let since order show enough prove since arf thus integer therefore follows thus implication trivial suppose let integers clear let suppose stable note since also therefore let integers since consequently thus sets leaps let numerical semigroup genus gaps remembering ordered pair called leap simply leap set leaps denoted tizziotti villanueva note authors work ordered pairs sparse numerical semigroups genus considering element pairs called single leap double leap respectively based idea positive integer ordered pair called simply numerical semigroups genus called single leap simply single leap called double leap simply double leap positive integer set denoted convenience define positive integer simplify notation denote cardinality set definition sets numerical semigroup get disjoint union observe numerical semigroup genus finite union disjoint sets next result let remember numerical semigroup called hyperelliptic theorem let numerical semigroup genus finite union sets mutually disjoint particular precisely hyperelliptic case integer case positive integer proof let numerical semigroup genus gaps let prove item theorem hyperelliptic integer hyperelliptic reciprocally otherwise single leap contradiction moreover since integer follows integer assertion follows first assertion item follows directly item prove last assertion theorem numerical semigroups integer thus positive integer hand integer implies positive integer proof complete proposition let numerical semigroup genus case positive integer positive integer proof prove definition positive integer implies suppose follows item follows directly previous item item clear integer reciprocally suppose item theorem follows suppose since unique double leap integer therefore conclude integer item trivial since positive integer theorem let numerical semigroup genus sparse numerical semigroup case positive integer sparse numerical semigroup frobenius number positive integer sparse numerical semigroup proof prove first suppose sparse numerical semigroup since converse implication follows directly prove equality true since frobenius number suppose hyperelliptic theorem item equality follows since theorem tizziotti villanueva item gaps since sparse theorem item note definitions follows therefore proving result finally prove item suppose sparse numerical semigroup item item implies using item converse implication follows directly item numerical semigroup let numerical semigroup genus theorem shows integer item theorem motivates define new class numerical semigroups generalize sparse numerical semigroups definition let integer positive numerical semigroup genus called numerical semigroup note numerical semigroup positive integer particular numerical semigroup called pure numerical semigroup definition numerical semigroup positive integer sparse numerical semigroups numerical semigroups also pure numerical semigroups sparse numerical semigroups genus positive example integer ordinary numerical semigroup numerical semigroup integer numerical semigroups definition condition equivalent another properties follows main result theorem let integer let numerical semigroup genus gaps conductor following statements equivalent numerical semigroups integer integer positive integer proof prove initially suppose numerical semigroup given exist thus suppose first let suppose exist since follows note interval exists exactly consecutive numbers thus integer since choice largest index property gap exist therefore contradiction thus prove implication suppose exist positive integer note contradiction item finally proof implication suppose number note since consecutive contradiction item therefore numerical semigroup corollary let integer let numerical semigroup conductor pure numerical semigroup numerical semigroup positive integer tizziotti villanueva proof initially suppose pure numerical semigroup thus numerical semigroup integer since reciprocally suppose positive integer theorem item follows follows pure numerical semigroup example let integer integer considerer numerical semigroup genus pure numerical semigroup fact suppose pure numerical interval gaps thus conductor follows integer semigroup since follows corollary suppose thus gaps integers integer therefore pure numerical semigroup structure numerical semigroups positive integer let collection numerical semigroups let collection pure numerical semigroups denote lemma sparse numerical semigroup proof prove first suppose contradiction reciprocally suppose let integer particular therefore proposition item prove suppose collection sparse numerical semigroups item follows theorem item reciprocally suppose let theorem item sparse numerical semigroup proof complete numerical semigroups following result generalization theorem theorem let integer let numerical semigroup genus numerical semigroup frobenius number mvm let numerical semigroup mvm proof prove lemma item equality true since frobenius number suppose thus hyperelliptic theorem item positive integer equality follows since mvm theorem item let gaps property since mvm definition follows result prove suppose item numerical semigroup mvm mvm converse implication clear let collection numerical semigroups theorem tizziotti villanueva sequence strictly ascending chain positive integers proof show item lemma clear theorem follows example show item follows directly definition pure numerical semigroup prove note let genus another case taking max follows case second equality obvious item trivial item lemma item item previous theorem allows define equivalence relation following way given interested frobenius varieties concept introduced rosales show families numerical semigroups examples frobenius varieties remember intersection two numerical semigroups numerical semigroup numerical semigroup genus also numerical semigroup see details definition frobenius variety nonempty set numerical semigroups fulfilling following conditions genus lemma let integer proof let genus respectively clear thus suppose let gaps gaps genus gaps note let gaps gaps gaps numerical semigroups since gaps gaps follows hand gaps follows therefore numerical semigroup proposition let integer set frobenius variety proof condition definition satisfied previous lemma show condition also satisfied fact let genus gaps genus equal gaps theorem item follows figure illustrates retrospective numerical semigroups generalization sparse arf numerical semigroups numerical semigroups pure numerical semigroups numerical semigroups pure numerical semigroups numerical semigroups sparse numerical semigroups arf numerical semigroups ordinary semigroups trivial numerical semigroup figure diagram numerical semigroups tizziotti villanueva references arf une interpretation suite des ordres branche proc london math barucci dobbs fontana maximality properties numerical semigroups applications analytically irreducible local domains mem amer math soc campillo munuera parameters codes related arf semigroups ieee trans inform theory contiero moreira veloso structure numerical sparse semigroups applications weierstrass points journal pure applied algebra munuera torres villanueva sparse numerical semigroups lecture notes computer science applied algebra algebraic algorithms codes berlin heilderberg oliveira weierstrass semigroups canonical ideal curves manuscripta rosales families numerical semigroups closed finite intersections frobenius number houston math rosales branco arf numerical semigroups journal algebra rosales numerical semigroups developments mathematics new york united states villanueva semigrupos fracamente arf pesos semigrupos thesis unicamp faculdade universidade federal naves brasil instituto exatas terra campus araguaia universidade federal mato grosso rodovia pontal araguaia brasil address guilherme address juan
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solvency markov decision processes interest taolue petr aistis faculty informatics masaryk university czech republic department computer science university oxford oct abstract solvency games introduced berger provide abstract framework modelling decisions investor whose goal avoid ever going broke study new variant model addition stochastic environment fixed increments decrements investor wealth introduce interest earned paid current level savings debt respectively study problems related minimum initial wealth sufficient avoid bankruptcy steady decrease wealth probability least present exponential time algorithm approximates minimum initial wealth show polynomial time approximation possible unless qualitative case show problem whether given number larger equal minimum initial wealth belongs conp show polynomial time algorithm would yield polynomial time algorithm games existence longstanding open problem also identify classes solvency mdps problem cases algorithms also give corresponding bankruptcy avoiding strategies acm subject classification probability statistics keywords phrases markov decision processes algorithms complexity market models introduction markov decision processes mdp standard model complex results decisions may random mdp set states state assigned set enabled actions every action determines distribution set successor states run starts state every step controller chooses enabled action process moves new state chosen randomly according distribution assigned action functions describe decisions controller called strategies may depend whole history computation choice actions may randomized mdps form natural model financial world model nuances financial markets various models developed see common property models actions correspond investment choices result typically random payoffs controller one common aims area find controller investor strives avoid undesirable events paper consider model based standard reward structures mdps closely related solvency games studied model designed captures essential properties investments assume mdps assign real reward every action collected whenever action chosen states supported czech science foundation grant forejt also affiliated brno czech republic licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany solvency markov decision processes interest mdp capture global situation market prices assets etc note usually plausible model prices stochastic process see rewards model money received positive rewards money spent negative rewards controller controllers compared ability collect reward finite infinite runs standard objectives total reward average reward suitable modelling behaviour investor allow temporary loss arbitrary amount money long sequence negative rewards undesirable normally controller access credit limited authors consider objective defined follows starting initial amount wealth step current wealth computed adding reward collected step goal find controller maximizes probability although model captures basic behaviour investor lacks one crucial aspect usually present financial environment interest interests model value received holding certain amount cash conversely cost negative balance accommodate interests propose following extension objective fix interest rate starting initial wealth step compute current wealth adding collected reward also interest economical motivation model controller earn additional amount wealth lending assets fixed interest conversely controller debt pay interest creditors clarity presentation suppose interest earned positive wealth interest paid debts hence objective manage wealth stays threshold keep decreasing negative infinity precisely want maximize probability lim inf intuitively lim inf means controller ultimately need borrow money lim inf means controller able sustain interest payments income lim inf controller sustain interest payments bankrupts important observation objective closely related another objective concerning discounted total reward concretely given discount factor discounted total reward accumulated run defined weighted sum rewards actions run weight action particular threshold problem asks maximize probability given threshold problem considered variant threshold problem problem asks given probability infimum threshold maximal probability discounted reward surpassing threshold least show every controller probability discount factor equal probability lim inf interest rate effectively shows interreducibility problems note interpretation discount factor inverse interest natural financial mathematics contribution introduce model solvency mdps interests referred solvency mdps brevity allows capture complex dynamics wealth management uncertainty show every solvency mdp bound notational convenience define interest rate number usual interest rate percentage money unit time taolue chen petr aistis simaitis wealth bound bankruptcy surely avoided matter controller another bound wealth bankruptcy inevitable nevertheless also show still might infinitely many reachable values wealth two bounds main results paper concentrate complexity computing minimal wealth controller stay away bankruptcy let infimum initial wealths starting state controller avoid bankruptcy lim inf probability least overall goal compute number solution problem important investor whose aim keep risk bankruptcy acceptable level first consider qualitative case case show connection games discounted total reward objectives using results show oblivious strategy one looks current state independent wealth accumulated far starting state wealth avoids bankruptcy probability one problem whether given encoded binary conp also obtain reduction discounted total reward games showing improving complexity bound might difficult addition number computed time follows restricted class solvency markov chains one enabled action every state value computed polynomial time main part paper concerns quantitative case arbitrary probability bound give algorithm approximates given absolute error actually show algorithm runs time polynomial number control states exponential log log log rmax interest rate rmax maximal reward associated action employing reduction knapsack problem show complexity lowered polynomial either log log log rmax unless give algorithm given initial wealth computes initial wealth increased probability avoiding bankruptcy least sup moreover via aforementioned interreducibility discounted solvency mdps establish new complexity bounds approximation discounted mdps note aforementioned algorithms employ careful rounding numbers representing current wealth choosing right precision rounding quite intricate step since naive choice would yield algorithm paper organized follows introducing necessary definitions clarifying relation discounted mdps section summarise results qualitative problem section section give contributions quantitative problem related work processes involving interests formal models naturally emerge field financial mathematics model financial market presented chapter every step investor allocate current wealth riskless bonds receives interest according fixed interest rate several risky stocks whose price subject random fluctuations optimization investor portfolio respect various utility measures studied however portfolio optimization problem considered case trading stops fixed number steps contrast concentrate stability investor wealth also model analysed mainly solvency markov decision processes interest mathematical perspective characterizing form optimal portfolios focus efficient algorithmic computation optimal investor behaviour issues stability algorithms considered related models concern total accumulated reward properties model especially close solvency games fact mdps single control state investor aims keep total accumulated reward energy games see two competing players stochastic behaviour mdps counter seen storage current value wealth models differ topic studied paper consider interest wealth makes fundamentally different terms properties setting set wealths reachable given initial wealth nontrivial limit points also three aforementioned models objective stay positive wealth focus different objective capture idea admissible debt long possible maintain debt limit mentioned work also related threshold discounted total reward objectives considered authors studied cases case particular gave algorithm compute probability careful analysis shows algorithm doublyexponential complexity planning horizon number steps process halts encoded binary proposed approximate probability discretisation wealth worst error approximation matter small discretisation step taken optimality equation characterising optimal probabilities provided case algorithm proposed moreover considered problem case giving approximation algorithm although consider mdps upper bound approximation lower bound easily carried finitehorizon case thus establish new complexity bounds approximation finite discounted mdps also mention introduced percentile performance criteria controller aims find strategy achieving specified value limit average reward specified probability level percentile preliminaries denote sets natural integer rational real numbers respectively index set member vector denote encoding size object denoted use log refer binary logarithm assume numbers represented binary rational numbers represented fractions integers assume familiarity basic notions probability theory given countable set use dist denote probability distributions definition mdp markov decision process mdp tuple countable set vertices finite set actions dist partial transition function assume every set actions available set actions defined nonempty denote succ support markov chain mdp one action per vertex taolue chen petr aistis simaitis given initial vertex mdp evolves follows infinite path run sequence succ finite path history prefix run ending vertex word form refer set runs runsm set histories histm finite infinite path denote finite path strategy function every history assigns distribution actions available last vertex strategy deterministic always assigns distributions choose action probability memoryless depends last vertex history use set strategies history histm determines set cone consisting runs prefix mdp vertex strategy associate probability space runsm generated cone unique probability measure every history cone otherwise empty product equal drop subscript mdp clear context definition solvency mdp solvency markov decision process tuple finite set states mdp partial gain function interest rate stipulate every value defined iff solvency markov chain solvency mdp one action per state configuration solvency mdp represented pair semantics given infinitestate mdp every define whenever sometimes distinguish refer strategies runs strategies runs intended strategy oblivious memoryless make decision based current wealth objectives given solvency mdp initial configuration interested set runs wealth always stays finite bound denoted win runsm runsm lim inf intuitively objective models ability investor bankrupt compensate incurred interest obtaining sufficient gains denote val win maximal probability winning given wealth inf val infimum wealth sufficient winning probability paper mainly interested problems computing approximating values also address problem computing convenient strategy investor given initial wealth precise definition mean convenient strategy given section theorem say strategy initial configuration win strategy called almost surely winning strategy win called almost surely losing example consider following solvency mdp solvency markov decision processes interest work profit invest loss work invest profit loss depicted arrows figure example invest function given bold numbers next actions work take extremely large value keep example computations simpler mdp models choices person either work ensures certain relatively small income invest larger amount money take significant risk starting configuration debt example strategy strategy always chooses work easily seen get win since constant gains high enough cover interest incurred debt optimal strategy pick work histories ending configuration pick invest otherwise strategy shows val suppose investor wants find wealth needed make sure probability winning least wants compute number equal see observe configuration optimal strategy must pick invest probability results debt impossible recover finally observe val since strategy always chooses work demonstrates function val given state may continuous relationship discounted mdps problems study solvency mdps closely related another decision making model called discounted mdps threshold objectives discounted mdp tuple first four components solvency mdp discount factor semantics discounted mdp given mdp reward function disc every run assigns total discounted reward disc threshold objective asks controller maximize given threshold probability event run disc consider solvency mdp initial configuration discounted mdp threshold objective note initial configuration fixed natural correspondence runs initiated runs initiated identify run run correspondence naturally extends strategies mdps assume mdps identical sets runs strategies proposition let win proof suffices show every run win disc fix run def define every disc empty sum assumed equal obviously every disc thus disc disc exists equal similarly disc disc infimum wealth along finite see appendix win follows many natural problems solvency mdps value computation etc polynomially equivalent similar natural problems discounted mdps threshold objectives particular problem interreducible taolue chen petr aistis simaitis problem discounted mdps aim supremum threshold suitable strategy probability risk discounted reward qualitative case section establish connection qualitative problem solvency mdps determining whether given state number problem determining winner discounted games definition discounted game finite discounted game tuple sets player states respectively initial state transition relation reward function discount factor strategy player discounted game function every strategy memoryless depends last state pair strategies players yields unique run run game given depending whether discounted total reward run defined disc discounted game problem asks given game value whether strategy player strategies player disc run strategy called winning proposition problem determining whether state solvency mdp interreducible polynomial time problem determining whether corresponding discounted mdp show latter discounted game problem let first fix discounted mdp say run realisable strategy idea reduction relies following lemma proved appendix lemma satisfies runs realisable using lemma construct game stipulating results actions chosen player instead chosen randomly vice versa technical details reduction presented appendix next theorem follows reduction fact memoryless deterministic strategies suffice discounted games theorem every solvency mdp exists oblivious deterministic strategy winning every configuration discounted game problem exists pseudopolynomial algorithm computing optimal value also one players controls states game problem solved polynomial time hence get following theorem theorem qualitative problem solvency mdps conp moreover pseudopolynomial algorithm computes every state actually use slightly different variants discounted game problem reductions discounted mdps problem respectively nevertheless establish desired complexity bounds solvency markov decision processes interest restricted class solvency markov chains compute decide qualitative problem done polynomial time note existence reduction games discounted games suggests improving complexity difficult since polynomialtime algorithm solvency mdps would give algorithm games existence longstanding open problem area graph games quantitative case section formulates results quantitative questions solvency mdps start proposition showing restrict attention subset since every state two values strategies winning losing respectively intuitively values represent wealth positive negative interest dominate gain function important consequence proposition combined deterministic strategies suffice maximize probability winning therefore rest section consider deterministic strategies proposition proved appendix proposition every state solvency mdp rational numbers def arg inf win def arg sup win encoding size polynomial computed polynomial time using linear programming techniques moreover win every strategy illustrate proposition return example note obviously every max example shows using restrict set interesting configurations trivial bounds def also define global versions bounds def accordance economic interpretation model call configuration form rentier configuration proposition follows every run visits rentier configuration belongs win note although proposition suggests restrict analysis configurations set reachable configurations bounds still infinite general following example shows example consider solvency mdp show configuration reachable exactly steps initial configuration satisfies hence reachable state space infinite numbers pairwise different set let reachable configuration form satisfying conditions one step reach configurations clearly otherwise would thus contradicts definition remains show one values satisfies conditions simple exercise give proof appendix note interest restricted integer reachable configuration space finite initial configuration holds taolue chen petr aistis simaitis hence reachable wealth multiple finitely many numbers means one use algorithms mdps minimising probability reach configuration however general case possible need devise new techniques approximation algorithms subsection show approximate algorithm depends following theorem allows certain sense explained soon approximate function val theorem algorithm computes solvency mdp initial configuration given rational number strategy val strategy configuration running time algorithm polynomial log min pmin min exponential log log rmax max prove theorem later first argue theorem important right consider following scenario suppose investor starts wealth plausible assume initial wealth strictly fixed instead one assume investor willing acquire small additional amount wealth represented exchange substantial benefit benefit consists fact small difference initial wealth allows investor compute execute strategy risk bankruptcy provably greater lowest risk achievable original wealth note strategy may val proceed theorem providing approximation theorem given solvency mdp state rational numbers possible approximate absolute error time polynomial log exponential log log pmin min rmax theorem proof suppose already know use algorithm theorem algorithm returns know otherwise conclude initially know order approximate absolute error suffices perform log iterations procedure finishing later show time complexity algorithm improved polynomial either log log unless proof theorem rest section fix solvency mdp initial configuration first establish existence strategy given small additional amount wealth reaches rentier configuration exponential number steps probability least val show compute strategy exponential time establish proof following proposition use suitable bellman functional whose unique fixed point equal proof found appendix solvency markov decision processes interest proposition every initial configuration every strategy starting ensures hitting rentier configuration log steps probability least val particular log win val previous proposition shows number strategy theorem computed examining possible behaviours first steps however since log close number exponential thus trivial algorithm unfolds mdp initial configuration tree depth tree computes strategy maximising probability reaching rentier configuration complexity key idea allowing reduce complexity round numbers representing wealth configurations numbers polynomial size size chosen carefully error introduced rounding large enough thwart computation following assume log log log since otherwise compute strategy number computing action maximizes probability reaching rentier configuration formalise notion rounding numbers appearing configurations let rational number say two configurations denoted one following conditions holds greater less equal clearly indeed equivalence set every member quotient set tuple form either interval length one intervals denote maximal element putting also denote equivalence class let proposition define mdp representing unfolding dag depth current wealth always rounded least integer multiple greater configurations exceeding upper dropping lower threshold proposition immediately recognized winning losing unfolded mdp formally defined follows definition unfolded mdp let solvency mdp two numbers define mdp transition function unique function satisfying following bounded interval distribution defined iff assigns every vertex self loop vertex every action given every action size well time needed construct log min denote hit set runs contain vertex form almost rentier set runs hit configuration form steps particular event hitting rentier configuration steps following lemma proved appendix shows adequately approximates behaviour lemma let arbitrary configuration following holds every hit taolue chen petr aistis simaitis input mdp state number output number strategy satisfying conditions theorem see proposition log log log initial configuration see definition hit val val theorem else compute see lemma return strategy see given algorithm algorithm approximating def hit supremum taken moreover number finite representation strategy computed time finish proof theorem let put easy computation shown appendix proves thanks assumption log log log proposition strategy val lemma get hit val part lemma compute time strategy number val words strategy reaches probability least configuration units wealth away rentier note initial configuration fixed strategy viewed function sequence states actions history since current wealth inferred sequence initial wealth suppose fix initial configuration instead keeping strategy use strategy selects action observing sequence states actions obvious reach rentier configuration probability least win required remains analyse complexity construction analysis merely technical postponed appendix thm results described section summarised form pseudocode algorithm lower bounds complement positive results given lower complexity bounds theorem problem deciding whether given furthermore existence following algorithms possible unless algorithm approximating absolute error time polynomial log log max exponential log min algorithm approximating absolute error time polynomial log min log exponential log numbers rmax pmin theorem proof sketch show construct given instance knapsack problem solvency mdp item values suitably encoded probabilities certain solvency markov decision processes interest transitions item weights encoded rewards associated actions show instance knapsack solution certain state certain number computed instance holds also show order decide inequality suffices constructed mdp approximate absolute error intuitively corresponds fact polynomial approximation algorithm knapsack achieve constant absolute error get part use slight modification approach let note crucial component aforementioned reductions val may continuous function see example intuitively allows recognise whether current wealth always encodes weight set items surpasses threshold proof found appendix note thanks interreducibility proposition suitably rephrased results theorems hold also approximation discounted mdps conclusions introduced solvency mdps model apt analysis systems interest paid received accumulated wealth analysed complexity fundamental problems proposed algorithms approximate minimum wealth needed win given probability compute strategy achieves goal obtained new results problem discounted mdps several important directions future study one question deserving attention find algorithm computing approximating val usual approaches discretising state space work case since function val continuous thus difficult bound error introduced discretisation another direction implementation algorithms evaluation references andersson miltersen complexity solving stochastic games graphs proceedings isaac pages berlin heidelberg rieder markov decision processes applications finance springer berger kapur schulman vazirani solvency games proceedings fst tcs volume lipics pages schloss dagstuhl boda filar time consistent dynamic risk measures math meth boda filar lin spanjers stochastic target hitting time problem early retirement ieee trans automat etessami approximating termination value mdps stochastic games inf chakrabarti alfaro henzinger stoelinga resource interfaces proc emsoft volume lncs pages heidelberg springer chatterjee doyen energy parity games proceedings icalp part volume lncs pages springer chatterjee doyen energy parity markov decision processes proceedings mfcs volume lncs pages springer chung sobel discounted mdps distribution functions exponential utility maximization siam contr taolue chen petr aistis simaitis filar krass ross percentile performance criteria limiting average markov decision processes ieee trans automat hamilton time series analysis volume cambridge univ press hull options futures derivatives pearson martin determinacy blackwell games symb logic markov decision processes finance dynamic options international series operations research management science sobel variance discounted markov decision processes appl white minimizing threshold probability discounted markov decision processes math anal lin minimizing risk models markov decision processes policies depending target values math anal zwick paterson complexity mean payoff games graphs tcs solvency markov decision processes interest technical appendix proofs section proof proposition show missing part proof proposition need show infimum wealth along finite follows fact every disc disc max max proof lemma lemma let strategy runs realisable proof suppose lemma hold let run realisable der disc let maxs maxs let let run form denoting disc disc however probability runs nonzero contradiction assumption interreducibility discounted mdps discounted games let first fix discounted mdp define game whenever defined whenever clarity presentation first assume rational number polynomial encoding size show get rid assumption let strategy define strategy player arbitrary action satisfying player strategies run corresponds run realisable disc run disc every run realisable disc run direction let winning player strategy assume memoryless define strategy assume let run realisable disc fix strategy player taolue chen petr aistis simaitis defined easily show disc run disc contradicts winning drop assumption rational number polynomial encoding size case represent number symbolically triplet minimal polynomial numbers represent fact interested positive one lying interval root consider aforementioned game represented surely determining winner game least hard determining winner standard discounted games since rational numbers also represented triplet form thus remains show problem determining winner conp let recall standard games see works first guesses winning memoryless deterministic strategy player verifies using linear programming techniques strategy player decrease discounted reward linear programs coefficients represented triplet form solved turing machine time polynomial encoding size triplets degree algebraic extension defined adjoining coefficients program see theorem linear program obtained guessing strategy contains one coefficient may irrational namely generates extension degree thus verify guessed strategy winning polynomial time conp upper bound proceed similarly direction interreducibility discount game define discounted mdp arbitrary function satisfying iff rest proof proceeds way remark rational number pseudopolynomial algorithm qualitative problem solvency mdps immediately obtained pseudopolynomial algorithm discounted games algorithm paper iterates pseudopolynomial number steps suitable bellman functional iteration performs polynomial number additions multiplications divisions comparisons involve discount reduction number since operations computed polynomial time algebraic numbers triplet form proposition intermediate results lie extension generated algorithm pseudopolynomial even games symbolically represented discounts note game optimal value resulting algorithm rational even irrational corresponds minimal threshold achievable probability discounted mdp rational discount rationality minimal threshold shown devising suitable bellman functional omit argument essential paper proofs section proof proposition present extended version proposition proposition every state solvency mdp rational numbers solvency markov decision processes interest def arg inf win def arg sup win encoding size polynomial solutions following linear programs max succ min succ moreover win every strategy proof first show actually real numbers let gmax max maximal gain occurs solvency mdp fix arbitrary gmax denoting gmax run starting form hence get take minimal gain gmin min proceed similarly proceed proving values satisfy optimality conditions first present proof value assume initial wealth follows action succ matter strategy picks actions following states accumulated wealth never falls know ensures strategy wins almost surely direction assume let construct strategy loses positive probability let pick action arg let arg action state bounding constraint follows strategy continues picking actions way ensures steps exists run ending state nonzero measure difference wealth equal value find wealth accumulated finite path nonzero probability implies wealth eventually reached thus strategy lose positive probability prove values satisfy optimality conditions first assume initial wealth satisfies let linear program know actions successors succ hence matter choice strategy next step difference wealth least successor show induction steps difference wealth least wealth going let initial wealth let construct strategy winning positive probability consider strategy picks action arg let taolue chen petr aistis simaitis arg action state bounding constraint hence follows strategy continues picking actions way ensures steps exists run ending state nonzero measure difference wealth equal value find wealth accumulated finite path nonzero probability implies wealth eventually reached thus strategy win positive probability bound encoding size numbers follows standard results linear programming supplement example distinguish two cases firstly put secondly argue allows put suppose opposite gives thus contradiction proof proposition let recall fact thanks proposition restricted deterministic strategies proposition every initial configuration every strategy starting ensures hitting rentier configuration log steps probability least val particular log win val proof proposition proceed series lemmas first show vector unique fixed point suitable bellman operator let state action number denote set vectors succ satisfy intuition behind vectors strategy picks action must vector succ probability winning successor form must least consider bellman operator defined space follows min inf max vectors lemma vector fixed point operator proof assume sake contradiction pick arbitrary denote side inequality definition follows succ words solvency markov decision processes interest starting strategy choose action first step second step current vertex succ switch strategy ensures winning probability least strategy must exist due using approach probability winning least last inequality holds since get contradiction definition remains show arbitrary fixed suffices show every strategy similarly previous paragraph must succ strategy chooses first step second step play configuration strategy winning probability least exists behave strategy second step onwards since follows indeed win denote set vectors satisfy also denote sup lemma also moreover every pair vectors proof let arbitrary assume sake contradiction definition strategy wins probability thus every end every succ time definition suitable combining two inequalities get contradiction inequality established similar way second part fix arbitrary vectors show let choose arbitrary definition follows def max def max assume case handled symmetric way max max second inequality follows facts putting together using fact obtain since chosen arbitrarily result follows taolue chen petr aistis simaitis previous lemma shows operator contraction mapping set equipped supremum norm banach space banach theorem unique fixed point must equal lemma moreover every vector sequence converges fixed point approaches infinity consider vector every lemma consider strategy starting strategy ensures hitting rentier configuration steps probability least proof proceed induction case trivial assume lemma holds consider must action vector max order reach rentier configuration steps probability least strategy proceed follows first step chooses aforementioned action second step play state probability current wealth greater induction strategy switch strategy reaches rentier configuration steps probability least follows strategy ensures reaching rentier configuration steps probability least finish proof proposition first inequality follows lemmas follows log holds log particular val val thus strategy chosen strategy lemma val proof lemma lemma let arbitrary configuration following holds every hit def sup hit supremum taken moreover number finite representation strategy computed time proof remind reader restricted deterministic strategies proposition purpose proof let define absorbing vertices vertices mdp form bounded interval moreover denote set histories last solvency markov decision processes interest vertex form previous vertices also denote set histories contain exactly one rentier configuration thus every proper prefix contain configuration note hit cone cone following assume initial configuration runs equal initial vertex runs equal first describe certain natural correspondence runs used throughout proof let set histories contain one absorbing vertex history exactly one history note viewed function histories injective hand every history unique history vertex either absorbing straightforward induction reveals wdi always round numbers nearest multiple follows hit cone function viewed mapping histories may injective use ker denote kernel denote equivalence class fix strategy define strategy follows every history contain absorbing vertex set histories reach absorbing vertex escape choose action easy verify every history cone cone follows hit cone cone cone cone cone first equality second line follows fact cone cone disjoint events prefix vice versa standard results mdps reachability objectives memoryless strategy hit hit maximum taken strategies thus suffices prove strategy satisfying hit let history let corresponding history prove induction every wdk case trivial since suppose equality actions considered taolue chen petr aistis simaitis wdi first inequality follows fact interval length containing first equality third line follows next inequality follows induction hypothesis consequence first inequality second line follows next inequality follows fact last vertex history form thus cone proceed similarly define strategy follows given history histories chooses arbitrary action note representation computed time polynomial since use standard algorithm mdps reachability objectives compute computed time linear length easy verify every history runs runs combining previous observations get cone cone cone hit last equality second line use fact injective proof log log last inequality holds assumed log log log actually computed online new configurations visited play solvency markov decision processes interest complexity analysis theorem conclude proof theorem previous observations complexity rewritten log min log min log noting log conclude complexity indeed polynomial log min exponential log log proof theorem theorem problem deciding whether given furthermore existence following algorithms possible unless algorithm approximating absolute error time polynomial log log exponential log max min algorithm approximating absolute error time polynomial log log exponential log min numbers rmax pmin theorem proof begin part second part similar give proof reduction knapsack problem let instance knapsack problem items assume weight value ith item respectively bound weight value items put knapsack respectively denote wtot vtot without loss generality assume numbers nonzero item weights integers restriction knapsack still vtot otherwise transform instance dividing numbers number vtot without influencing existence solution show compute time polynomial encoding size knapsack instance solvency mdp interest rate number solution instance knapsack distinguished state interest rate chosen way inequality holds first put next set define transitions follows note indeed defines probability distribution since vtot rewards defined following way wtot taolue chen petr aistis simaitis figure solvency mdp constructed simple knapsack instance items achieve greater compactness picture omits loops action states action rewarded states wtot ensures wtot run visits winning current wealth upon entering least wtot rewards zero figure illustrates construction simple example finally set clearly mdp number computed polynomial time note would imply vtot case knapsack admit solution thus assume strategy state reached probability shown induction since hence get state reached probability probability run contains strategy picks state otherwise let choose deterministic strategy arbitrary initial configuration obviously interpret choice action choosing item packed knapsack denote set indexes chooses action set items chosen packed knapsack straightforward induction reveals conditional reaching state current wealth upon reaching state two consequences first run reaches one states current wealth always bounded wtot thus run starting winning reaches losing reaches second solvency markov decision processes interest run reaches state happens probability current wealth upon reaching state equal thus conditional probability winning condition play reaches items total weight least wtot words items total weight otherwise conditional probability since assume item weights integers aforementioned probability iff items contained total weight claim strategy set solution original instance knapsack set items selected inclusion knapsack satisfies usual constraints weight value win first suppose solution knapsack instance since total weight items follows play reaches happens probability current wealth least wtot conditional reaching probability winning moreover know every run reaches winning well know happens probability conclude using strategy win probability least hand solution original instance two cases consider either wtot case probability winning vtot conditional probability winning upon reaching way win reach case probability winning strictly smaller probability reaching conditional probability winning plus probability visiting either case probability winning less follows instance knapsack admits solution win words iff val every recognize whether case suffices approximate value absolute error furthermore constructed mdp log log poly wtot polynomial poly thus algorithm approximating time polynomial log min log would imply existence algorithm knapsack second part theorem proof almost divide rewards sufficiently large number forcing rmax polynomial encoding size knapsack instance show order decide whether thus whether instance admits solution suffices approximate absolute error wtot sum weights items instance formally difference gain function constructed mdp put ensures run hits winning current wealth upon entering least rewards zero taolue chen petr aistis simaitis easy verify states always winning losing respectively given strategy current wealth upon entering equal initial wealth let number interval strategy conditional probability winning condition reached iff wtot otherwise since conditional probability wtot iff contains items weight since item weights integers argue exactly way previous part solution knapsack instance iff ensures winning probability least follows knapsack instance solution possible win probability least initial wealth check whether case suffices approximate absolute error constructed mdp log log poly polynomial poly since rmax thus approximation algorithm running time polynomial log min log exponential log would yield algorithm knapsack
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dataset simulator data pose estimation visual odometry slam author nov elias henri guillermo tobi davide abstract new vision sensors dynamic vision sensor davis incorporate conventional globalshutter camera sensor pixel array sensors great potential robotics computer vision allow combine benefits conventional cameras sensors low latency high temporal resolution high dynamic range however new algorithms required exploit sensor characteristics cope unconventional output consists stream asynchronous brightness changes called events synchronous grayscale frames purpose present release collection datasets captured davis variety synthetic real environments hope motivate research new algorithms robotics applications addition intensity images asynchronous events provide inertial measurements camera poses system latter allows comparing pose accuracy egomotion estimation algorithms quantitatively data released standard text files binary files rosbag paper provides overview available data describes simulator release create synthetic data keywords cameras visual odometry slam simulation dataset website datasets simulator found web http video containing visualizations datasets https requirements properties enable design new class algorithms robotics standard cameras typically ideal motion blur image saturation high latency however way cameras convey information completely different traditional sensors paradigm shift needed deal introduction past fifty years research devoted standard cameras rolling global shutter cameras last years cameras successfully used commercial autonomous mobile robots cars drones vacuum cleaners mention despite recent progress believe advent cameras revolutionize robot sensing landscape indeed performance mobile robot tasks navigation depends accuracy latency perception latency depends frequency sensor data plus time takes process data typical current pipelines latencies order puts hard bound maximum agility platform camera virtually eliminates latency data transmitted using events latency order another advantage cameras high dynamic range standard cameras makes ideal scenes characterized large illumination changes key properties cameras related datasets exist two recent datasets also use davis rueckauer delbruck barranco first work tailored comparison optical flow estimation algorithms rueckauer delbruck contains synthetic real datasets pure rotational degrees freedom dof motion simple scenes strong visual contrasts ground truth acquired using inertial measurement unit imu robotics perception group university zurich switzerland neuroinformatics university zurich eth zurich switzerland institute corresponding author elias mueggler robotics perception group university zurich switzerland mueggler work supported starting grant darpa fla program google faculty research award qualcomm innovation fellowship national centre competence research robotics nccr swiss national science foundation uzh forschungskredit grant davis sensor axes definitions figure adapted delbruck visualization event output davis blue dots mark individual asynchronous events polarity events shown figure davis camera visualization output contrast datasets contain arbitrary motion variety artificial natural scenes precise camera poses system similar work barranco focus create dataset facilitates comparison methods visual navigation tasks end ground robot equipped davis microsoft kinect sensor davis mounted unit thus could excited scene contains checkerboards books chair ground truth acquired encoders unit ground robot wheel odometry therefore subject drift contrast dataset contains motion variety scenes precise camera poses system subject drift davis sensor dynamic vision sensor davis brandli see fig event camera transmits events addition frames events changes detected continuous time events timestamped resolution transmitted asynchronously time occur event tuple pixel coordinates event timestamp event polarity event sign brightness change representation sometimes also referred representation aer davis spatial resolution pixels visualization event output shown fig events frames generated physical pixels hence spatial offset events frames due low latency high temporal resolution range cameras promising sensors mobile robot applications since event cameras brightness changes transmitted redundant data transmitted required bandwidth thus depends motion speed type scene additional advantage robotic applications high dynamic range compared expensive cameras allows indoor outdoor operation without changing parameters since pixels independent large contrast changes also take place within scene course last seven years several groups including demonstrated use eventbased sensors variety tasks slam weikersdorfer kueng kim rebecq optical flow cook benosman bardow visual odometry censi scaramuzza localization robotics mueggler line detection localization yuan ramalingam reconstruction rebecq image reconstruction mosaicing kim reinbacher orientation estimation gallego scaramuzza trajectory estimation mueggler however methods evaluated different specific datasets therefore compared datasets propose tailored allow comparison pose tracking visual odometry slam algorithms since cameras davis currently still expensive usd data also allow researchers without equipment use data research davis imu addition visual output events frames davis includes imu provides gyroscope accelerometer data thus enabling design algorithms davis camera imu mounted directly behind centered image sensor pixel array center distance imu shares nearly position event sensor photoreceptor optical center camera since lens dependent calibration discussed page imu axes aligned visual sensor axes see fig specifically imu video illustration https dataset simulator log two samples corresponding intensity jump larger contrast threshold provided events perfect measurements sampling quantization condition image reconstructed event stream point time accumulating events huk according log log samples log actual events predicted events figure davis simulator event generation using piecewise linear time interpolation intensities given rendered images simplicity images rendered fixed rate invensense integrates gyroscope measure range accelerometer range integrates six adcs digitizing gyroscope accelerometer outputs khz sample rate davis simulator simulation offers good baseline working new sensors davis based operation principle ideal davis pixel created simulator given virtual scene trajectory moving davis within generates corresponding stream events intensity frames depth maps used computer graphics software generate thousands rendered images along specified trajectory ensuring motion consecutive images smaller pixel pixel keep track time last event triggered location map timestamps also called surface active events benosman combined time interpolation rendered image intensities allows determining brightness changes predefined amount given contrast threshold time images thus effectively providing continuous timestamps events generated asynchronously time interpolation additional benefit solves problem generate millions images second sequence would required deliver timestamps absence interpolation specifically fig illustrates operation simulator single pixel continuous intensity signal pixel log black sampled times rendered images blue markers samples used determine times events data linearly interpolated consecutive samples crossings resulting lines red levels given multiples contrast threshold horizontal lines specify timestamps events red dots observed simple interpolation scheme allows higher resolution event time stamps rendered images generation multiple events rendered image time selects pixel updated every event pixel updated time used scheme check reconstructed image agreed rendered image several points time specifically intensity error confined quantization interval event generation operates brightness pixels computed rendered color images using recommendation luma according formula rgb channels linear color space better resemble operation davis realistic event noise extremely difficult model due complex behavior event sensors respect bias settings factors provided simulation datasets include event noise nevertheless simulator datasets created useful tool prototyping new algorithms implementation available datasets section describe datasets provide datasets contain asynchronous event stream intensity images inertial measurements gyroscope accelerometer khz camera poses systemk precision indoor datasets intrinsic camera matrix information comes precise timestamps datasets captured outside system office outdoors ground truth provided datasets collected using motorized linear slider ground truth collected using slider position due vibrations induced slider motor noisy imu data recorded data format datasets provided standard text form described convenience also imu data sheet https https https https davis simulator use optitrack system naturalpoint shapes wall poster boxes outdoors dynamic calibration office urban motorized linear slider motorized slider hdr motorized slider objects synthetic planes synthetic walls figure dataset scenes dataset simulator file description one event per line one image reference per line images referenced one measurement per line one measurement per line camera parameters line content timestamp polarity timestamp filename timestamp timestamp table description dataset format downloaded binary rosbag files details website format text files described table pose respect arbitrary origin pointing upwards orientation provided unit quaternion scalar vector components respectively convention proposed standard jpl breckenridge values reported units timestamps originally recorded posix subtracted lowest timestamp offset datasets start zero helps avoid numerical difficulties dealing microsecond resolution timestamps events images provided png files list images timestamps provided separate file typical framerate varies exposure time imu axes orientation optical coordinate frame positive aligned optical axis list datasets provided datasets summarized table fig datasets contain increasing speeds different scenes varying degrees shapes poster boxes datasets motion first starts excitation single degree freedom separately combined faster excitations performed leads increasing difficulty higher event rate time hdr sequences hdr poster hdr boxes slider hdr spotlight used create large intrascene contrasts hdr poster measured dark bright areas respectively outdoors datasets acquired urban environment walking running ground truth available returned precisely location large loop dynamic datasets collected office environment viewed system moving person first sitting desk moving around calibration dataset also available instance case user wishes use different camera model different methods calibration dimensions calibration pattern checkerboard tiles lower half table different settings lenses focus etc used thus provide intrinsic calibration calibration datasets available slider close slider far slider hdr close slider hdr far datasets thk tek tcek tcej standard notation notation xbi thk tek tcej figure calibration coordinate frames transformations involved case device two different positions red loop two stations device used first type calibration problems form xbi blue loop used second type calibration problems form use combination approaches solve constant calibration transform recorded motorized linear slider parallel textured wall respectively datasets applied two different sets biases parameters davis listed table first set labeled indoors used datasets outdoors walking outdoors running urban set outdoors applied simulated datasets used contrast threshold simulation simulation respectively simulated scenes also provide world model blender fig addition intensity images events datasets include depth map image frame encoded values openexr data format calibration first calibrated davis intrinsically using checkerboard pattern computed calibration applied subsequent dataset recordings poses provide event camera eye davis moved hand dominant motion described name shapes rotation shapes translation shapes poster rotation poster translation poster boxes rotation boxes translation boxes hdr poster hdr boxes outdoors walking outdoors running dynamic rotation dynamic translation dynamic calibration office zigzag office spiral urban slider close slider far slider hdr close slider hdr far slider depth simulation simulation motion rotation incr speed translation incr speed dof incr speed rotation incr speed translation incr speed dof incr speed rotation incr speed translation incr speed dof incr speed dof incr speed dof incr speed dof walking dof running rotation incr speed translation incr speed dof incr speed dof slow zigzag slow spiral slow linear slow linear const speed linear const speed linear const speed linear const speed linear const speed translation circle dof scene fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig fig yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes table list datasets note calibration dataset applies upper half table datasets use different lenses calibrations ground truth duration maximum translation speed maximum rotational speed number events start end pose large loop ground truth motorized linear slider imu data due vibrations simulated davis using blender imu data included bias diffbn offbn onbn prbp prsfbp refrbp indoors coarse fine outdoors coarse fine table list biases applied davis davis uses two stages biases coarse fine report trackable hand attached camera also included calibration dataset case different camera model improved calibration method required intrinsic camera calibration used standard pinhole camera model radialtangential distortion using implementation ros used three radial distortion coefficients two tangential distortion distortion coefficients listed order opencv provide dataset postcalibration case another method preferred calibration indoor datasets provide accurate highfrequency pose data system however coordinate frame used system different optical coordinate frame davis thus performed calibration acquiring datasets fig shows coordinate frames transformations used solve calibration problem frames world hand camera fig checkerboard transformations fig shows compact standard notation calibration problems explicit one euclidean transformation tba homogeneous matrix representation maps points frame frame according tba specifically first use linear algorithm tsai lenz provide initial solution handeye calibration problem xbi correspondences relative thk tek poses different times respectively unknown transformation using second formulation eye calibration problems form http dataset simulator tcej transformations pose respectively refined jointly estimating transformations minimize reprojection error image plane min xmn xmn squared euclidean distance measured projection xmn checkerboard corner camera predicted corner function intrinsic camera parameters extrinsic parameters predicted using motioncapture data problem solved iteratively using method initial value given provided abovementioned linear algorithm included dataset postcalibration case another method preferred pose gives position orientation event camera respect world motioncapture system hence already incorporates computed transformation system outputs apply calibration thj directly report thj pose calibration calibration dataset used compute euclidean transformation camera imu reference frames running publicly available software kalibr furgale calibration dataset provides transformation camera eye imu given timu rotation matrix approximately identity camera imu axes orientation translation dominantly along optical axis imu couple behind camera optical center used lens additionally due imu filter imu measurements lag behind images events temporal shift also reported kalibr known issues clock drift offset clocks system davis observed clock drift counteract clock drift reset clocks dataset recording since datasets rather short order min effect drift negligible small offset davis timestamps present since timestamps reset software references bardow davison leutenegger simultaneous optical flow intensity estimation event camera ieee int conf computer vision pattern recognition cvpr barranco fermuller aloimonos delbruck dataset visual navigation neuromorphic methods frontiers neuroscience benosman clercq lagorce ieng bartolozzi visual flow ieee trans neural netw learn syst brandli berner yang liu delbruck latency global shutter spatiotemporal vision sensor ieee circuits breckenridge quaternions proposed standard conventions technical report nasa jet propulsion laboratory censi scaramuzza visual odometry ieee int conf robotics automation icra cook gugelmann jug krautz steger interacting maps fast visual interpretation int joint conf neural networks ijcnn delbruck villanueva longinotti integration dynamic vision sensor inertial measurement unit electronically stabilized vision int conf circuits systems iscas furgale rehder siegwart unified temporal spatial calibration systems int conf intelligent robots systems iros gallego scaramuzza accurate angular velocity estimation event camera ieee robotics automation letters kim handa benosman ieng davison simultaneous mosaicing tracking event camera british machine vision conf bmvc kim leutenegger davison reconstruction tracking event camera eur conf computer vision eccv kueng mueggler gallego scaramuzza lowlatency visual odometry using feature tracks int conf intelligent robots systems iros mueggler gallego scaramuzza continuoustime trajectory estimation vision sensors robotics science systems rss mueggler huber scaramuzza pose tracking maneuvers int conf intelligent robots systems iros rebecq gallego scaramuzza emvs eventbased stereo british machine vision conf bmvc rebecq gallego scaramuzza evo geometric approach parallel tracking mapping ieee robotics automation letters reinbacher graber pock intensityimage reconstruction event cameras using manifold regularisation british machine vision conf bmvc rueckauer delbruck evaluation algorithms optical flow inertial measurement sensor frontiers neuroscience tsai lenz new technique fully autonomous efficient robotics calibration ieee trans robotics weikersdorfer hoffmann conradt simultaneous localization mapping vision systems int conf computer vision systems icvs yuan ramalingam fast localization tracking using event sensors ieee int conf robotics automation icra
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deep health care text classification vinayakumar barathi ganesh anand kumar soman center computational engineering networking cen amrita school engineering coimbatore amrita vishwa vidyapeetham amrita university india anandkumar soman oct abstract health related social media mining valuable apparatus early recognition diverse antagonistic medicinal conditions mostly existing methods based machine learning learning working note presents recurrent neural network rnn long memory lstm based embedding automatic health text classification social media mining task two systems built classify tweet tweet level rnn lstm used extracting features activation function last layer facilitates distinguish tweets different categories experiments conducted social media mining health applications shared task amia experiment results considerable however proposed method appropriate health text classification primarily due reason rely feature engineering mechanisms introduction expansion micro blogging platforms twitter internet progressively utilized spread health information instead similarly wellspring twitter allows users share status messages typically called tweets restricted characters time tweets expresses opinions topics thus analysis tweets considered significant task many applications health related applications health text classification taken account special case text classification existing methods used machine learning methods feature engineering commonly used features tags term document frequency semantic features mentions chemical substance disease wordnet synsets adverse drug reaction lexicon proposed ensemble based approach classifying adverse drug reactions tweets recently deep learning methods performed used many tasks mainly due rely feature engineering mechanism however performance deep learning methods implicitly relies large amount raw data sets make use unlabeled proposed approach based convolutional neural network adverse drug event detection though data sets task task limited paper proposes rnn lstm based embedding method background hyper parameter selection section discusses concepts tweet representation deep learning algorithms particularly recurrent neural network rnn long memory lstm mathematical way tweet representation representation tweets typically called tweet encoding contains two steps tweets tokenized words first step moreover words transformed second step dictionary formed assigning unique key word tweet unknown words tweet assigned default key retain word order tweet word replaced unique number according dictionary tweet vector sequence made length choosing particular length tweet sequences long particular length discarded short padded zeros type word vector representation passed input word embedding layer task maximum tweet sequence length thus train matrix shape valid matrix shape passed input embedding layer task maximum tweet sequence length thus train matrix shape valid matrix shape passed input embedding layer word embedding layer transforms word vector word embedding using following mathematical operation input shape weights word embedding words word embedding dimension denotes number top words denotes number unique words denotes size word embedding vector kind mathematical operation transforms discrete number vectors continuous numbers word embedding layer captures semantic meaning tweet sequence mapping high dimensional geometric space high dimensional geometric space called embedding space embedding properly learnt semantics tweet encoding real valued vectors similar tweets appear cluster close high dimensional geometric space select optimal parameter embedding size two trails experiments run embedding size experiment learning rate set experiment embedding size performed well rnn lstm networks thus rest experiments embedding size set embedding layer output vector passed rnn variant lstm layer rnn lstm obtain optimal feature representation feature representation passed dropout layer dropout layer contains removes neurons connections randomly acts regularization parameter task output layer contains sigmoid activation function sof tmax activation function task recurrent neural network rnn variant recurrent neural network rnn enhanced model feed forward network ffn introduced input sequences arbitrary length passed rnn transition function maps hidden state vector recursively hidden state vector calculated based transition function present input sequence previous hidden state vector mathematically formulated follows hit otherwise kind transition function results vanishing exploding gradient issue alleviate lstm lstm network contains special unit typically called memory block memory block composed memory cell set gating functions input gate forget gate output gate control states memory cell transition function lstm units defined igt wig pig qig big ogt wog pog qog bog tanh gti igt hit ogt tanh input time step weight parameters sigmoid activation function denotes multiplication experiments section discusses data set details task task followed experiments related parameter tuning task aims classifying twitter posts either existence adverse drug reaction adr task aims classifying twitter posts personal medication intake possible medication intake data sets two tasks provided shared task committee detailed statistics reported table table task data set composed train validation test data sets table task data statistics data training validation testing total tweets total classes adr mentioned tweets adr mentioned tweets table task data statistics data training validation testing total tweets total classes personal medicine intake possible medicine intake non intake results experiments trained using backpropogation time bptt graphics processing unit gpu enabled computational framework conjunction keras framework ubuntu submitted one run based lstm task two runs composed one run based rnn one based lstm task evaluation results given shared task committee reported table table task results run adr precision adr recall adr table task results run precision classes recall classes classes conclusion social media mining considerably important source information many health applications working note presents rnn lstm based embedding system social media health text classification due limited number tweets performance proposed method less however obtained results considerable open way future apply social media health text classification moreover performance lstm based embedding task good comparison task primarily due fact target classes task data set imbalanced hence proposed method applied large number tweets corpus order attain best performance references lee kathy ankit agrawal alok choudhary disease surveillance using twitter data demonstration flu cancer proceedings acm sigkdd international conference knowledge discovery data mining kdd pages new york usa acm lee kathy ankit agrawal alok choudhary mining social media streams improve public health allergy surveillance international conference advances social networks analysis mining asonam pages aug sarker abeed graciela gonzalez portable automatic text classification adverse drug reaction detection via training journal biomedical informatics jonnagaddala toni rose jue dai binary classification twitter posts adverse drug reactions proceedings social media mining shared task workshop pacific symposium biocomputing big island usa dai musa touray jitendra jonnagaddala shabbir feature engineering recognizing adverse drug reactions twitter posts information ravikumar komandur elayavilli yue hongfang liu detecting signals noisy ensemble classifiers help identify adverse drug reaction tweets proceedings social media mining shared task workshop pacific symposium biocomputing zhang zhifei nie xuyao zhang ensemble method binary classification adverse drug reactions social media proceedings social media mining shared task workshop pacific symposium biocomputing nguyen huy nguyen deep neural architecture sentiment classification twitter social networking arxiv preprint lee kathy ashequl qadir sadid hasan vivek datla aaditya prakash joey liu oladimeji farri adverse drug event detection tweets convolutional neural networks proceedings international conference world wide web international world wide web conferences steering committee elman jeffrey lfinding structure time cognitive science hochreiter sepp urgen schmidhuber long memory neural computation gers felix jrgen schmidhuber fred cummins learning forget continual prediction lstm gers felix nicol schraudolph jrgen schmidhuber learning precise timing lstm recurrent networks journal machine learning research werbos paul backpropagation time proceedings ieee abadi barham chen chen davis dean devin ghemawat irving isard kudlur tensorflow system machine learning osdi vol barathi ganesh anand kumar soman distributional semantic representation health care text
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cnn hyperspectral image classification hyungtae sungmin heesung booz allen hamilton mclean virginia army research laboratory adelphi maryland jan lee hyungtae eum sungmin abstract paper address dataset scarcity issue hyperspectral image classification thousands pixels available training difficult effectively learn convolutional neural networks cnns cope problem propose novel cnn containing shared parameters across multiple hyperspectral datasets network also contains portions designed handle datasetspecific spectral characteristics associated classification tasks approach first attempt learn cnn multiple hyperspectral datasets fashion moreover experimentally shown proposed network trained three widely used datasets outperform baseline networks trained single dataset index hyperspectral image classification convolutional neural network cnn shared network cross domain domain adaptation introduction introduction convolutional neural network cnn brought forth unprecedented performance increase classification problems many different domains including rgb rgbd hyperspectral images performance increase made possible due ability cnn able learn express deep wide connection input output using huge number parameters order learn huge set parameters large scale dataset become significant requirement size given dataset insufficient learn network one may consider using larger external dataset better learn large set parameters instance girshick introduced domain adaptation approach network trained large scale source domain imagenet dataset finetuned target domain object detection dataset applying cnn hyperspectral image classification problem also face similar issue hyperspectral datasets classification results classification layers hyperspectral analysis layers hyperspectral analysis layers hyperspectral analysis layers shared convolutional layers classification layers classification layers hyperspectral classification cnn optimized fashion fig cnn portions cnn hyperspectral analysis layers classification layers handling classification tasks shared convolutional layers generate common information applies datasets note shared portions cnn optimized fashion large scale hyperspectral dataset available typical hyperspectral image classification data contains pixels small portion pixels used training order tackle data scarcity issue need way make use multiple hyperspectral dataset domain several challenges arise devising approach applied multiple hyperspectral domains first hyperspectral datasets contain different wavelength range spectral reflectance bands furthermore applying domain adaptation approach infeasible large scale auxiliary dataset hyperspectral image classification available therefore developed novel cnn architecture simultaneously performs network learning classification multiple hyperspectral datasets architecture consists three components hyperspectral analysis layers shared convolutional layers classification layers portion hyperspectral analysis layers present analyze information hyperspectral analysis layers end built classification layers performs classification different dataset two components connected shared convolutional layers learns common information across domains three components optimized fashion information acquired multiple datasets fed layers concurrently training leads better learning shared convolutional layers via dataset augmentation overall cnn architecture depicted figure paper used three mostly used hyperspectral datasets indian pines salinas university pavia demonstrate effectiveness crossdomain cnn experimental results show novel architecture outperforms baseline networks one dataset used train network consistently terms classification accuracy contributions approach listed classification layers shared convolutional layers first attempt learn cnn multiple hyperspectral datasets novel cnn optimized fashion conv conv conv conv conv max pool relu relu relu proposed method conv max pool figure shows proposed cnn architecture backbone cnn architecture stream proposed architecture used modified version hyperspectral image classification cnn introduced lee kwon modification carried adding batch normalization layer convolutional layer removing local response normalization layers introducing layers bypass process data normalization training testing backbone cnn contains sequentially connected filter bank one convolutional layer two residual modules three convolutional layers filter bank consists filters analyze characteristics residual module includes two convolutional layers cnn architecture filter bank stream responsible analyzing characteristics assigned hyperspectral analysis layers last three convolutional layers function classification layers remaining layers middle architecture consist second convolutional layer two residual modules assigned crossdomain shared convolution layers softmax loss consistent classification accuracy increase datasets conv conv conv conv conv relu relu relu relu relu conv convolution layer batch normalization layer max pool max polling layer relu rectified linear unit layer dropout layer softmax loss softmax loss layer shared layers fig architecture red blue green blocks indicate portion cnn black ones shared across streams different datasets detailed specifications architecture denoted last green stream indicate number spectral bands number classes respectively different datasets evaluation evaluation settings used three hyperspectral datasets indian pines salinas university pavia experiments indian pines dataset includes pixels spectral reflectance bands cover range spatial resolution indian pines dataset classes classes relatively large numbers samples used salinas dataset contains classes pixels spectral bands high spatial resolution salinas dataset shares frequency characteristics indian pines dataset sensor aviris used data acquisition university pavia dataset provides classes pixels spectral bands cover spectral range spatial table selected classes evaluation numbers training test samples indian pines class woods total training test class brocooli green weeds brocooli green weeds fallow fallow rough plow fallow smooth stubble celery grapes untrained soil vineyard develop corn senesced green weeds lettuce romaines lettuce romaines lettuce romaines lettuce romaines vineyard untrained vineyard vertical trellis total table classification accuracy dataset university pavia salinas training test class asphalt meadows gravel trees sheets bare soils bitumen bricks shadows total training test domains brings notable classification accuracy improvements compared individually trained cases individual cnn cnn gain indian pines salinas university pavia references saurabh gupta ross girshick pablo jitendra malik learning rich features images object detection segmentation eccv resolution salinas dataset university pavia dataset use classes datasets contain classes relatively small number samples dataset randomly partitioned train test sets according table ross girshick jeff donahue trevor darrell jitendra malik convolutional networks accurate object detection segmentation ieee transactions pattern analysis machine intelligence vol classification accuracy shown table proposed cnn outperforms individual networks indian pines salinas university pavia respectively note individual networks trained without shared portion network kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition cvpr conclusion paper introduced novel cnn concurrently learn perform hyperspectral image classification multiple datasets shared portion network trained using multiple hyperspectral datasets proposed approach effective optimizing high capacity cnn cases single dataset domain used approach first attempt exploit multiple hyperspectral datasets training cnn fashion experimentally demonstrated using shared layers across hyungtae lee heesung kwon contextual deep cnn based hyperspectral classification igarss christian szegedy wei liu yangqing jia pierre sermanet scott reed dragomir anguelov dumitru erhan vincent vanhoucke andrew rabinovich going deeper convolutions cvpr
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published conference paper iclr earning epresent rograms raphs marc brockschmidt microsoft research cambridge mabrocks feb miltiadis allamanis microsoft research cambridge miallama mahmoud simon fraser university vancouver canada mkhademi bstract learning tasks source code formal languages considered recently work tried transfer natural language methods capitalize unique opportunities offered code known sematics example dependencies induced using variable function distant locations often considered propose use graphs represent syntactic semantic structure code use deep learning methods learn reason program structures work present construct graphs source code scale gated graph neural networks training large graphs evaluate method two tasks var naming network attempts predict name variable given usage var isuse network learns reason selecting correct variable used given program location comparison methods use less structured program representations shows advantages modeling known structure suggests models learn infer meaningful names solve var isuse task many cases additionally testing showed var isuse identifies number bugs mature projects ntroduction advent large repositories source code well scalable machine learning methods naturally leads idea big code largely unsupervised methods support software engineers generalizing existing source code allamanis currently existing deep learning models source code capture shallow textual structure sequence tokens hindle raychev allamanis parse trees maddison tarlow bielik flat dependency networks variables raychev models miss opportunity capitalize rich semantics source code work take step alleviate including two additional signal sources source code data flow type hierarchies encoding programs graphs edges represent syntactic relationships token well semantic relationships variable last formal parameter argument called stream key insight exposing semantics explicitly structured input machine learning model lessens requirements amounts training data model capacity training regime allows solve tasks beyond current state art explore two tasks illustrate advantages exposing semantic structure programs first consider var naming task allamanis raychev given source code correct variable name inferred sequence subtokens requires understanding variable used requires reasoning lines code far work done intern microsoft research cambridge published conference paper iclr var root clazz var recclass clazz string name name figure snippet detected bug ravendb project code slightly simplified model detects correctly variable used highlighted yellow slot incorrect instead first placed slot reported problem fixed apart source file secondly introduce variable misuse prediction task var isuse network aims infer variable used program location illustrate task figure shows slightly simplified snippet bug model detected popular project specifically instead variable clazz variable first used yellow highlighted slot existing static analysis methods detect issues even though software engineer would easily identify error experience achieve high accuracy tasks need learn representations program semantics tasks need learn semantic role variable counter filename additionally var isuse learning variable usage semantics filename needed required fill blank element task related methods learning distributed representations natural language words mikolov glove pennington however learn much richer structure data flow information work step towards learning program representations expect valuable wide range tasks code completion variable looking advanced bug finding lock using object summarize contributions define var isuse task challenge machine learning modeling source code requires learn semantics programs section present deep learning models solving var naming var isuse tasks modeling code graph structure learning program representations graphs section iii evaluate models large dataset million lines source code showing best model achieves accuracy var naming task accuracy var isuse task beating simpler baselines section document practical relevance var isuse summarizing bugs found mature software projects subsection implementation graph neural networks simpler task found https elated ork work builds upon recent field using machine learning source code artifacts allamanis example hindle bhoopchand model code sequence tokens maddison tarlow raychev model syntax tree structure code works language models code find predicting variable method identifiers one biggest challenges task closest work work allamanis learn distributed representations variables using usages predict names however use data flow information aware model raychev bichsel use conditional random fields model variety relationships variables ast elements types predict variable names types resp deobfuscate android apps without considering flow data explicitly works variable usages deterministically known beforehand code complete remains unmodified allamanis work remotely related work program synthesis using sketches automated code transplantation barr however approaches require set published conference paper iclr specifications examples test suites complete gaps rather statistics learned big code approaches thought complementary since learn statistically complete gaps without need specifications learning common variable usage patterns code neural networks graphs gori defferrard kipf welling gilmer adapt variety deep learning methods input used series applications link prediction classification grover leskovec semantic role labeling nlp marcheggiani titov somewhat related source code work wang learn representations mathematical formulas premise selection theorem proving var isuse task detecting variable misuses code task requires understanding reasoning program semantics successfully tackle task one needs infer role function program elements understand relate example given program fig task automatically detect marked use clazz mistake first used instead task resembles standard code completion differs significantly scope purpose considering variable identifiers mostly complete program task description view source code file sequence tokens tokens variables furthermore let refer set variables scope location variables used without raising compiler error call token want predict correct variable usage slot define separate task slot given correctly select training evaluation purposes correct solution one simply matches ground truth note practice several possible assignments could considered correct several variables refer value memory odel rograms raphs section discuss transform program source code program graphs learn representations program graphs encode program text also semantic information obtained using standard compiler tools gated graph neural networks work builds gated graph neural networks ggnn summarize graph composed set nodes node features list directed edge sets number edge types annotate vector representing features node embedding string label node associate every node state vector initialized node label sizes state vector feature vector typically use larger state vectors padding node features propagate information throughout graph messages type sent neighbors message computed current state vector arbitrary function choose linear layer case computing messages graph edges time states updated time particular new state node computed aggregating incoming messages edge type aggregation function implement elementwise summation given aggregated message current state vector node state next time step computed gru gru recurrent cell function gated recurrent unit gru cho dynamics defined equations repeated fixed number time steps use state vectors last time step node graph convolutional networks gcn kipf welling schlichtkrull would simpler replacement ggnns correspond special case ggnns gated recurrent units published conference paper iclr expressionstatement invocationexpression memberaccessexpression assert notnull argumentlist data flow edges foo indices added clarity red dotted lastuse edges green dashed lastwrite edges dashdotted purple computedfrom edges simplified syntax graph line fig blue rounded boxes syntax nodes black rectangular boxes syntax tokens blue edges child edges double black edges nexttoken edges figure examples graph edges used program representation program graphs represent program source code graphs use different edge types model syntactic semantic relationships different tokens backbone program graph program abstract syntax tree ast consisting syntax nodes corresponding nonterminals programming language grammar syntax tokens corresponding terminals label syntax nodes name nonterminal program grammar whereas syntax tokens labeled string represent use child edges connect nodes according ast induce order children syntax node additionally add nexttoken edges connecting syntax token successor example shown fig capture flow control data program add additional edges connecting different uses updates syntax tokens corresponding variables token let set syntax tokens variable could used last set may contain several nodes example using variable conditional used branches even syntax tokens follow program code case loops similarly let set syntax tokens variable last written using add lastread resp lastwrite edges connecting elements resp additionally whenever observe assignment expr connect variable tokens occurring expr using computedfrom edges example semantic edges shown fig extend graph chain uses variable using lastlexicaluse edges independent data flow else link two occurrences also connect return tokens method declaration using returnsto edges creates shortcut name type inspired rice connect arguments method calls formal parameters matched formalargname edges observe call foo bar method declaration foo inputstream stream connect bar token stream token finally connect every token corresponding variable enclosing guard expressions use variable guardedby guardedbynegation edges example else add guardedby edge resp guardedbynegation edge ast node corresponding finally types edges introduce respective backwards edges transposing adjacency matrix doubling number edges edge types backwards edges help propagating information faster across ggnn make model expressive leveraging variable type information assume statically typed language source code compiled thus variable known type use define learnable embedding function known types additionally define ype types also leverage rich type hierarchy available many languages map variable type set supertypes implements type compute type representation used state updates number propagation steps per ggnn layer fixed instead several layers used experiments gcns generalized less well ggnns published conference paper iclr variable maximum chose maximum natural pooling operation representing partial ordering relations type lattices using types allows generalize unseen types implement common supertypes interfaces example list multiple concrete types list int list string nevertheless types implement common interface ilist share common characteristics training randomly select subset ensures training known types lattice acts like dropout mechanism allows learn good representation types type lattice initial node representation compute initial node state combine information textual representation token type concretely split name node representing token subtokens classtypes split two subtokens class types camelcase average embeddings subtokens retrieve embedding node name finally concatenate learned type representation computed discussed earlier node name representation pass linear layer obtain initial representations node graph programs graphs var aming given program existing variable build program graph discussed replace variable name corresponding variable tokens special slot token predict name use initial node labels computed concatenation learnable token embeddings type embeddings discussed run ggnn propagation time compute variable usage representation averaging representations slot tokens representation used initial state gru predicts target name sequence subtokens name inputstreambuffer treated sequence input stream buffer train architecture using maximum likelihood objective section report accuracy predicting exact name score predicting subtokens program graphs var isuse model var isuse program graphs need modify graph first compute context representation slot want predict used variable insert new node slot position corresponding hole point connect remaining graph using applicable edges depend chosen variable slot everything lastuse lastwrite lastlexicaluse guardedby edges compute usage representation candidate variable target slot insert candidate node connect graph inserting lastuse lastwrite lastlexicaluse edges would used variable used slot candidate nodes represents speculative placement variable within scope using initial node representations concatenated extra bit set one candidate nodes run ggnn propagation time context usage representation final node states nodes slot finally correct variable usage location computed arg maxv linear layer uses concatenation train using objective mplementation using ggnns sets large diverse graphs requires engineering effort efficient batching hard presence diverse shapes important observation large graphs normally sparse thus representation edges adjacency list would usually advantageous reduce memory consumption case easily implemented using sparse tensor representation allowing large batch sizes exploit parallelism modern gpus efficiently second key insight represent batch graphs one large graph many disconnected components requires appropriate make node identities unique makes batch construction somewhat found useful prepare minibatches separate thread tensorflow abadi implementation scales graphs per second training graphs per second using single nvidia geforce found fewer steps insufficient good results propagation steps help substantially published conference paper iclr gtx titan graphs average median nodes median edges ggnn unrolling iterations edge types forward backward edges original edge types size hidden layer set number types edges ggnn contribute proportionally running time example ggnn run ablation study using two common edge types nexttoken child achieves training test time hyperparameters generic implementation ggnns available https using simpler demonstration task valuation dataset collected dataset var isuse task open source projects github select projects picked projects github filtered projects could easily compile full using require compilation extract precise type information code including types present external libraries final dataset contains projects diverse set domains compilers databases million lines code full table shown appendix task detecting variable misuses collect data projects selecting variable usage locations filtering variable declarations least one replacement variable scope task infer correct variable originally existed location thus construction least one replacement variable picking would raise error type checking test datasets slot average alternative variables median dataset selected two projects development set rest projects selected three projects nseen roj est allow testing projects completely unknown structure types split remaining projects sets proportion splitting along files examples one source file set call test set obtained like een roj est baselines var isuse consider two bidirectional baselines local model simple bidirectional gru run tokens target location baseline set slot representation computed rnn usage context variable embedding name type variable computed way initial node labels ggnn baseline allows evaluate important usage context information task flat dataflow model avg rnn extension usage representation computed using another bidirectional rnn run tokens usage averaging computed representations variable token local context identical avg rnn significantly stronger baseline already takes structural information account averaging variables usages helps dependencies models pick variable maximizes var naming replace avg lbl uses model left right context tokens variable usage averages context representations corresponds model allamanis also test avg rnn var naming essentially replaces context model bidirectional rnn uantitative valuation table shows evaluation results models captures little information performs relatively badly avg lbl avg rnn capture information many variable usage sites explicitly encode rich structure problem still lag behind ggnn wide margin performance difference larger var isuse since structure semantics code far important within setting http sect additionally shows roc curves ggnn model var isuse task published conference paper iclr table evaluation models een roj est refers test set containing projects files training set nseen roj est refers projects files training data results averaged two runs var isuse accuracy auc var aming accuracy een roj est avg lbl avg rnn ggnn nseen roj est avg lbl avg rnn ggnn table ablation study ggnn model een roj est two tasks accuracy var isuse var naming ablation description standard model reported table nexttoken child lastuse lastwrite edges semantic edges nexttoken child syntax edges nexttoken child node labels tokens instead subtokens node labels disabled generalization new projects generalizing across diverse set source code projects different domains important challenge machine learning repeat evaluation using nseen roj est set stemming projects files training set right side table shows models still achieve good performance although slightly lower compared een roj est expected since type lattice mostly unknown nseen roj est believe dominant problem applying trained model unknown project domain fact type hierarchy unknown used vocabulary variables method class names etc differ substantially ablation study study effect design choices models run additional experiments show results table first varied edges used program graph find restricting model syntactic information large impact performance tasks whereas restricting semantic edges seems mostly impact performance var isuse similarly computedfrom formalargname returnsto edges give small boost var isuse greatly improve performance var naming evidenced experiments node label representation syntax node token names seem matter little var isuse naturally great impact var naming ualitative valuation figure illustrates predictions ggnn makes sample test snippet snippet recursively searches global directives file gradually descending root folder reasoning correct variable usages hard even humans ggnn correctly predicts variable usages locations except two slot software engineer writing code imaginable may make mistake misusing one variable place another since variables string variables type errors raised probabilities fig suggest potential variable misuses flagged model yielding valuable warnings software engineers additional samples comments found appendix furthermore appendix shows samples pairs code snippets share similar representations computed cosine similarity usage representation ggnn reader notice network learns group variable usages share semantic similarities together example checking null use variable yields similar distributed representations across code segments sample appendix published conference paper iclr bool tryfindglobaldirectivesfile string basedirectory string fullpath string path var directivesdirectory null return true path null return false figure var isuse predictions slots within snippet een roj est set servicestack project additional visualizations available appendix underlined tokens correct tokens model select among number string variables slot represent kind path ggnn accurately predicts correct variable usage slots reasoning complex ways variables interact among public int var var arraysegment byte readbytes int length size length buffer ensuretempbuffer length used read buffer size figure bug found yellow ravendb project code unnecessarily ensures buffer size length rather size model predicts correct variable iscovered variable isuse ugs used var isuse model identify likely locations bugs ravendb document database roslyn microsoft compiler framework manually reviewed sample top locations projects model confident choosing variable differing ground truth found three bugs projects figs show issues discovered ravendb bug fig possibly caused easily caught traditional methods compiler warn unused variables since first used virtually nobody would write test testing another isvalidbackup backupfilename false output error backuplocation look like valid backup throw new invalidoperationexception backuplocation look like valid backup figure bug found yellow ravendb project although backupfilename found invalid isvalidbackup user notified backuplocation invalid instead published conference paper iclr test fig shows issue although critical lead increased memory consumption fig shows another issue arising error message privately reported three additional bugs roslyn developers fixed issues meantime https one reported bugs could cause crash visual studio using certain roslyn features finding issues widely released tested code suggests model useful software development process complementing classic program analysis tools example one usage scenario would guide code reviewing process locations var isuse model identified unusual use prior focus testing expensive code analysis efforts iscussion onclusions although source code well understood studied within disciplines programming language research relatively new domain deep learning presents novel opportunities compared textual perceptual data local semantics rich additional information extracted using efficient program analyses hand integrating wealth structured information poses interesting challenge var isuse task exposes opportunities going beyond simpler tasks code completion consider first proxy core challenge learning meaning source code requires probabilistically refine standard information included type systems eferences abadi ashish agarwal paul barham eugene brevdo zhifeng chen craig citro greg corrado andy davis jeffrey dean matthieu devin tensorflow machine learning heterogeneous distributed systems arxiv preprint miltiadis allamanis earl barr christian bird charles sutton learning natural coding conventions foundations software engineering fse miltiadis allamanis earl barr christian bird charles sutton suggesting accurate method class names foundations software engineering fse miltiadis allamanis hao peng charles sutton convolutional attention network extreme summarization source code international conference machine learning icml miltiadis allamanis earl barr premkumar devanbu charles sutton survey machine learning big code naturalness arxiv preprint earl barr mark harman yue jia alexandru marginean justyna petke automated software transplantation international symposium software testing analysis issta bessey ken block ben chelf andy chou bryan fulton seth hallem charles asya kamsky scott mcpeak dawson engler billion lines code later using static analysis find bugs real world communications acm avishkar bhoopchand tim earl barr sebastian riedel learning python code suggestion sparse pointer network arxiv preprint benjamin bichsel veselin raychev petar tsankov martin vechev statistical deobfuscation android applications conference computer communications security ccs pavol bielik veselin raychev martin vechev phog probabilistic model code international conference machine learning icml kyunghyun cho bart van dzmitry bahdanau yoshua bengio properties neural machine translation approaches syntax semantics structure statistical translation published conference paper iclr defferrard xavier bresson pierre vandergheynst convolutional neural networks graphs fast localized spectral filtering neural information processing systems nips justin gilmer samuel schoenholz patrick riley oriol vinyals george dahl neural message passing quantum chemistry arxiv preprint marco gori gabriele monfardini franco scarselli new model learning graph domains ieee international joint conference neural networks ijcnn ieee aditya grover jure leskovec scalable feature learning networks international conference knowledge discovery data mining sigkdd acm abram hindle earl barr zhendong mark gabel premkumar devanbu naturalness software international conference software engineering icse thomas kipf max welling classification graph convolutional networks arxiv preprint yujia daniel tarlow marc brockschmidt richard zemel gated graph sequence neural networks international conference learning representations iclr chris maddison daniel tarlow structured generative models natural source code international conference machine learning icml diego marcheggiani ivan titov encoding sentences graph convolutional networks semantic role labeling acl tomas mikolov ilya sutskever kai chen greg corrado jeff dean distributed representations words phrases compositionality neural information processing systems nips jeffrey pennington richard socher christopher manning glove global vectors word representation emnlp veselin raychev martin vechev eran yahav code completion statistical language models programming languages design implementation pldi veselin raychev martin vechev andreas krause predicting program properties big code principles programming languages popl veselin raychev pavol bielik martin vechev probabilistic model code decision trees programming systems languages applications oopsla andrew rice edward aftandilian ciera jaspan emily johnston michael pradel yulissa detecting argument selection defects proceedings acm programming languages oopsla michael schlichtkrull thomas kipf peter bloem rianne van den berg ivan titov max welling modeling relational data graph convolutional network arxiv preprint armando program synthesis sketching university california berkeley mingzhe wang yihe tang jian wang jia deng premise selection theorem proving deep graph embedding advances neural information processing systems precision true positive rate published conference paper iclr seenprojtest unseenprojtest false positive rate curve seenprojtest unseenprojtest recall receiver operating characteristic roc curve figure roc curves ggnn model var isuse note axis starts table performance ggnn model var isuse per number candidate variables compute performance full ggnn model uses subtokens candidates accuracy een roj est accuracy nseen roj est erformance urves figure show roc curves ggnn model reader may observe setting false positive rate get true positive een roj est unseen test suggests model practically used high precision setting acceptable performance var isuse rediction amples list set samples een roj est projects comments model performance code comments formatting may altered typesetting reasons ground truth choice underlined sample var port return startingfrom endingat port startingfrom endingat endingat startingfrom port port startingfrom endingat port startingfrom endingat port startingfrom endingat model correctly predicts variables loop false positive rate widely accepted industry maximum acceptable limit bessey published conference paper iclr sample var path createfilename return name path name path name string variables confused semantic role inferred correctly sample global public void mergefrom input uint tag tag switch tag default break case input break payload model commonly confused aliases variables point location memory sample either choice would yielded identical behavior sample public override bool isdisposed get lock return hasobservers returnsto edge help predict variables otherwise would impossible published conference paper iclr sample summary notifies subscribed observers exception param error exception send public override void onerror exception error null throw new argumentnullexception nameof var default iobserver lock checkdisposed immutablelist iobserver true null foreach var error error hasobservers hasobservers error error error model predicts correct variables slots apart last reasoning last one requires interprocedural understanding code across class file published conference paper iclr sample private bool becomingcommand object message receivecommand return true else return false return true message response message message response message response message probe askedfordelete response message model predicts correctly usages except one slot reasoning snippet requires additional semantic information intent code sample var response resultsfilter typeof tresponse request httpmethod absoluteurl username useragent absoluteurl httpmethod username useragent model knows selecting correct string parameters matches formal parameter names sample throw new invalidoperationexception maxerror maxerror hard model reason conditionals especially rare constants slot published conference paper iclr earest eighbor ggnn sage epresentations show pairs nearest neighbors based cosine similarity learned representations slot marked dark blue usages marked yellow variablename set examples showing good bad examples brief description follows pair sample public void moveshapeup baseshape shape shape null int shapes shape return lock lockobject unobservableexceptionhanler null return false unobservableexceptionhanler handler slots checked similar representations sample public iactorref resolveactorref actorpath actorpath hasaddress actorpath return rootguardian actorpath actorpath actorpath tryparsecachedpath path actorpath hasaddress actorpath actorpath return rootguarding slots follow similar api protocols similar representations note function hasaddress local function seen testset published conference paper iclr sample foreach var filter filter public void adding elements object yields similar representations dataset collected dataset characteristics listed table full dataset set projects parsed json become available online table projects dataset ordered alphabetically kloc measures number lines code projects marked dev used development set projects marked dataset rest projects split dataset contains total name git sha klocs slots vars automapper benchmarkdotnet botbuilder choco dapper entityframework hangfire nancy ninject nlog opserver optikey orleans polly quartznet ravendbdev restsharp scriptcs servicestack sharex signalr wox description concurrent distributed framework mapping library benchmarking library sdk building bots windows package manager command line parser markdown parser object mapper library mapper background job processing library string manipulation formatting algorithmic trading engine http service framework json library code injection library logging library monitoring system assistive keyboard distributed virtual actor model resilience transient fault handling library scheduler document database rest http api client library reactive language extensions text editor web framework sharing application push notification framework application launcher
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queueing analysis block fading rayleigh channels low snr regime dec yunquan pingyi mails fpy mail department electronic engineering tsinghua university beijing china national mobile communication research laboratory southeast university china fading channels suffer channel fadings additive white gaussian noise awgn result impossible fading channels support constant rate data stream without using buffers paper consider information transmission block rayleigh fading channel low ratio snr regime characterize transmission capability channel terms stationary queue length distribution packet delay well data rate based memoryless property service provided channel block formulate transmission process discrete time discrete state queueing problem obtained results provide full characterization block rayleigh fading channels extended transmissions index rayleigh fading channel communication queueing analysis queue length distribution packet delay ntroduction wireless communication found applications recent years cellular network wlans satellite communication high speed railway communications unlike traditional wireline communications wireless communications suffer channel fading besides gaussian noise leads fluctuation instantaneous channel capacity brings great difficulty evaluation utilization wireless channels therefore characterizing kind service fading channel provide key challenge wireless communications transmission capability fading channel often characterized ergodic capacity outage capacity particular ergodic capacity statistical average value instantaneous capacity specifies maximum transmission rate fading channel large time scale capacity maximum achievable transmission rate outage probability constraint clear capacity focuses worst case reliability communications part channel service would wasted periods channel condition good therefore enough describe capability channels using data rate end authors proposed link layer channel model termed effective capacity jointly consider parameters traffic rate queue delay violations note effective capacity proposed based large deviation theory accurate large buffers delays fact wireless communications queueing theory connected naturally first due fluctuation instantaneous channel capacity buffer must used transmitter match source traffic channel transmission capability typical queueing problem second buffers commonly used engineering practice recently gallager berry discussed optimal power allocation policy delay constraints aided wireless communication system paper consider buffer aided communication block rayleigh fading channels low snr regime using tools queueing theory nutshell characterize kind service fading channels provide examine channel constant rate data input buffer describe channel transmission capability aspects queue length distribution packet delay although assumed buffer infinitely long results readily extended case however challenges applying queueing theory problem note channel gain varies block block fading channels thus time discrete fading communications moreover channel gain positive real number ranging zero infinity therefore input output process fading channels would discrete time continuous state markov process little references exists end one resort techniques stochastic process method transforming continuous state space discrete state space using quantization paper developed another method transform discrete time continuous states markov process discrete time discrete state markov chain key idea behind formulation memoryless property exponential distributions specific one block service provided rayleigh fading channel follows negative exponentially distribution although part service ability block consumed previous packet remaining service capability block follows distribution seen service ability provided new block based observation denote service time packet integer part actual service time thus state space queue length discrete epochs beginning blocks also discrete original discrete time continuous state queueing process transformed discrete time discrete state markov chain based model investigate stationary distribution queue length packet delay closed forms paper rest paper organized follows channel model queueing model formulation presented section stationary queue length distribution investigated iii proved stationary distribution queueing process departure epochs arbitrary epochs average packet delay obtained section obtained result also presented via numerical results section finally concluded work section ystem odel ueueing ormulation block fading channel model consider communication blockfading rayleigh channel additive white gaussian noise awgn channel channel gain stays fixed block varies independently different blocks let block length time varying channel gain block corresponding power gain probability density functions given respectively let denote transmit power limited transmit bandwidth noise power spectral density let denote path loss exponent distance transmitter receiver respectively instantaneous capacity block rayleigh fading channel nats service provided fading channel one block amount service provided fading channel successive blocks expressed also convenience notation define average received snr culculation get cumulative distribution cdf instantaneous capacity reduce low snr scenario considered paper cdf service one block given seen follows negative exponential distribution follows gamma distribution whose given gamma function markov chain model transmission capability time varying fading channel examined constant rate data stream infinite length first first fifo buffer used transmitter side match source traffic stream time varying channel service block let length queue buffer nats start block assume data served packets packet size equals traffic arriving buffer block rtb use denote time packet arriving block spend queue seen queueing process discrete time continuous state markov process unfortunately results available processes thus transformations needed construct discrete time discrete state queue define service time packet integer part actual service time number complete blocks period packet served firstly probability packet served within one block derived follows similarly fsk probability packet served blocks obtained fsk dxk holds independent importantly assumed hold sense memoryless property negative exponential distributions given following lemma lemma negative exponential distributed random variable memoryless namely therefore say service time packet packet actually completed blocks shown case part service capability block consumed continue service next packet queue shown channel service block low snr region follows negative exponential distribution memoryless lemma therefore remaining service capability block seen service provided new block sense packet consumes blocks similarly packet completed within one block defined service time zero way problem information transmission block fading rayleigh channel transformed classical discrete time queueing problem arrival process unit service time unite block poisson distributed random variable whose probability generating function pgf given throughout paper assumed queue stable information transmission channel formulated late arrival system queueing model immediate access packet arrives end block leaves beginning block service packet also assumed start beginning block service time packet zero assumed leave queue immediately beginning block following arrival let number packets queue arbitrary time usually markov chain define departure epoch packet service time packet zero must see either empty buffer start service new packet aftereffectless point queue length process let number packets buffer departure packet markov chain since arrival process transition probability matrix pgf service time given solving equation one get property pgf lim thus theorem proved stationary queue length arbitrary epochs subsection deals stationary queue length arbitrary epochs firstly let introduce two lemmas used lemma discrete random variable integers proof queue stable according classical queueing theory holds proof lemma follows stationary queue length departure epochs let limitation queue length process vector stationary queue length departure epochs theorem pgf stationary queue length departure epochs given iii ueue ength istribution section perform detailed investigation queue length process terms stationary queue length distribution multiplying sides equation take sum one lemma discrete random variable integers pgf proof next stationary queue length arbitrary epochs specified following theorem theorem stationary queue length distribution departure epochs arbitrary epochs proof denote limit distribution queue length process arbitrary epochs theorem proved number packets buffer let last departure epoch stays unchanged every boundary points blocks two adjacent departure epochs matter much packets arrives period define stays state let sojourn time buffer sojourn time equals service time head line packet buffer however buffer empty must wait one block arrival next packet first average stationary sojourn time sense probability distribution holds according theory renewal process define elapsed sojourn time state since know distribution law equals assume packets left buffer recall departure last packet queue length time means packets arrived interior every natural number differentiable derivative follows lemma completes proof stationary queue length distribution stationary distribution queue length process obtained pgf theorem know following discussion let introduce one lemma used lemma cauchy integral formula extended let simple closed positively oriented piecewise smooth curve domain let function analytic neighborhood interior every stationary queue length distribution turns follows circle follows lemma analytic define particularly multiplying sides taking sum one get seen function two singular points namely lower branch lambert function particularly one movable singularity since finite therefore within circle considered analytic according theory integral along closed curve invariable let circles centered origin radiuses since convergent within unit circle stationary expressed packet elay packet delay defined time interval arrival packet departure firstly packet delay consists service time packet arrives seeing buffer must wait waiting time service finally formulation paper another piece time packet spends system vestige time paper means service packet finished block according memoryless property negative exponential distribution although part service ability block consumed considered brand new block however packet still spend part block system called vestige time thus know next cdf theorem fifo discipline average packet delay given proof since one packet arrives block average time packet stays system equals average queue length little law lim investigate vestige time define new random variables firstly define service provided channel one block total amount service provided successive blocks definition remaining part packet blocks transmission positive using positive condition get assuming channel service next block remaining amount data needs blocks transmitted however vestige time remaining data transmitted within block turns see particularly cdf fuk upper gamma function therefore cdf lower gamma function since channel service next block random variables independent cdf derived follows fvk xsn dfu dfs average dxdu case define snp specifically one packet completed within one block vestige time equals actual service time cdf respectively get expected value finally whole probability formula expected value vestige time dxdu result complete proof combing umerical esults section provide numerical results illustrate stationary distribution packet delay infinite buffer model component variance assumed system bandwidth khz transmitting power dbw suppose distance transmitter receiver pathloss exponent block length chosen according definition tpb besides equivalent awgn capacity channel low snr region stationary distribution model vestige time average packet delay unit blocks fig average packet delay model corresponds left corresponds linear right fig stationary queue length distribution model curve presented linear left right therefore also seen ratio traffic rate awgn capacity seen fig average delay grows quickly particularly average delay corresponds left average vestige time corresponds linear right blocks seen average vestige time also increasing never exceeds stationary queue length distribution infinitebuffer model shown fig presented linear clear larger slower decreases easy understand heavier traffic leads longer queues importantly queue length grows probability occurs decreases approximately exponentially caire shamai shitz achievable throughput antenna gaussian broadcast channel ieee trans inf theory vol jul avestimehr tse outage capacity fading relay channel low snr regime ieee trans inf theory vol apr goldsmith capacity optimal resource allocation fading broadcast channels part outage capacity ieee trans inf theory vol mar negi effective capacity wireless link model support quality service ieee trans wireless vol july berry gallager communication fading channels delay constraints ieee trans inf theory vol may dong fan letaief murch rate information transmission fading channels ieee international wireless commun mobile comput conf limassol cyprusaugust dong fan letaief murch performance analysis communication block rayleigh fading channels queue length distribution overflow probability rate wireless commun mobile online published fan stochastic process theory applications press tsinghua university april jiaotong university advanced mathimatics lab complex variable functions press advanced education onclusion modern communications requires wireless channels provide qos guaranteed services characterize make full use service capability fading channels urgent problem paper studied problem block rayleigh fading channels using memoryless property block services however general fading channels answers quite unclear needs lot efforts acknowledgement work supported china major state basic research development program program national natural science foundation china nsfc nsfc open research fund national mobile communication research laboratory southeast university china eferences mceliece stark channels block interference ieee trans inf theory vol jan
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interference mitigation techniques coexistence mmwave users incumbents ghz jan ghaith hattab student member ieee eugene visotsky member ieee mark cudak member ieee amitava ghosh fellow ieee abstract millimeter wave spectra potential endow new radio mobile connectivity gigabit rates however pressing issue presence incumbent systems bands primarily fixed stations fss paper first identify key properties incumbents parsing databases existing stations major cities devise several modeling guidelines characterize deployment geometry antenna specifications second develop detailed interference framework compute aggregate interference outdoor users fss present several case studies dense populated areas using actual incumbent databases building layouts simulation results demonstrate promising coexistence majority fss experience interference well noise floor thanks propagation losses bands deployment geometry incumbent systems fss may incur higher interference propose several passive interference mitigation techniques exclusion zones spatial power control simulations results show techniques effectively protect fss without tangible degradation coverage index terms paper presented part ieee global communications conference globecom singapore december hattab nokia bell labs currently department electrical engineering university california usa visotsky cudak ghosh nokia bell labs arlington heights usa email ghattab coexistence interference management spectrum sharing mmwave wireless backhaul ntroduction new radio envisioned first cellular standard millimeter wave mmwave spectrum access paradigm shift towards mmwave access necessary scale explosive growth mobile traffic provide unparalleled network capacity peak data rates reaching tens gbps indeed mmwave spectrum attracted significant attention standard bodies industry academic community culminating recently federal communications commission fcc opened licensed spectrum cellular services specifically nevertheless still additional licensed spectra left future consideration candidate bands mmwave mobile networks advantages using bands also known twofold first band easily provide contiguous high bandwidth contrast provides maximum respectively second available worldwide enabling economies scale universal adoption common mmwave devices equally important operating higher end mmwave spectrum significantly different operating channel models increase path loss compensated using array larger number antenna elements addition several prototypes shown feasibility mmwave systems instance nokia huawei already demonstrated experimental systems designed operate ghz one key challenge using bands presence existing incumbents primarily fixed stations fss provide services wireless backhaul per fcc regulations incumbents must protected harmful interference thus objective study feasibility coexistence systems existing fss develop interference mitigation techniques ensure harmonious spectrum sharing related work several works studied spectrum sharing paradigms mmwave networks however works solely focused sharing among different mobile operators sharing frequency channels infrastructure etc spectrum sharing systems services recently attracted attention instance work focus coexistence radar systems whereas work studies coexistence wifi aforementioned works limited mmwave access paradigm also spurred interest coexistence studies example work study feasibility coexistence incumbents satellite systems coexistence fixed service relevant work paper one studies coexistence fss however work makes several modeling assumptions single assumed exist fixed distance system fixed portion links assumed nlos addition work focuses downlink interference mitigate uplink interference probing device proposed installed report excessive interference system work however focus passive interference mitigation techniques propose techniques require coordination system incumbents require probing devices contributions main contributions paper summarized follows characterizing incumbent analyze databases exiting fss four major areas united states characterize deployment geometry fss key antenna specifications analysis provides key insights feasibility coexistence well provide benchmarks accurate modeling fss interest academic community uplink interference analysis present detailed interference analysis framework compute aggregate uplink interference users fss also present random models user azimuth elevation antenna directions help reduce simulator complexity without degrading simulation accuracy passive interference mitigation propose several passive interference techniques require coordination system incumbent systems specifically propose exclusion zones base stations gnbs switch certain beams protect victim receivers techniques shown effective affect downlink coverage thus propose spatial power control defining quiet beams associated users transmit lower power discuss implementation techniques coexistence feasibility effectiveness proposed mitigation techniques validated via three case studies deploy systems dense urban suburban areas studies use databases existing fss actual building layouts accurate interference analysis results shown majority fss protected harmful interference due high propagation losses high attenuation due misalignment user antenna boresight deployment geometry fss systems fss experience higher interference proposed mitigation techniques provide significant protection effective switching gnbs vicinity fss finally simulation validate performance networks show distribution beams used gnb user making design insights mobile operators vendors paper organization rest paper organized follows system model presented section study fss deployment interference analysis framework presented section iii section respectively proposed mitigation techniques discussed section simulation results presented section conclusions drawn section vii ystem odel base stations gnbs consider deployment gnbs one deployed street corner height distance isd every site approximately disd site consists four sectors sector covers area simulation gnbs initialized grid fixed isd disd location gnb grid moved vicinity initial location ensure lies street corner resembling actual deployment gnbs txrus covers sector fig illustrative example gnb site sector equipped antenna array size antenna array assumed tilted downward angle antenna element gain transmit power apart nearest antenna element gnb site illustrated fig users consider outdoors user equipment terminals ues randomly deployed space fss outdoors attenuation due penetration losses indoor ues high equipped antenna array size antenna element gain transmit power respectively array height assumed titled upward angle also assumed two panels two sectors one covering thus user sense beams directions one panel active user beam association cell selection association measures received power reference signals sent different beams gnbs vicinity connects beam highest received incumbent fixed stations consider fss operate bands currently registered fcc database incumbents required database operating beam association algorithms criteria considered iterative beam association gnb antenna pattern steering azimuth angles degrees gnb antenna pattern steering azimuith angles degrees antenna pattern steering antenna gain dbi antenna gain dbi antenna pattern steering antenna gain dbi antenna gain dbi elevation angles degrees gnb beams elevation angles degrees beams fig different beams used dimension bands thus exact locations used similarly extract antenna specifications beamwidth gain azimuth orientation tilt different fss may operate different center frequencies aforementioned bands assume paper share spectrum system worst case scenario antenna patterns beam association data communications gnb use one available beams assume number beams per dimension twice number antennas dimension azimuth elevation beam pattern beamwidth approximately assume parabolic element pattern normalized azimuth elevation attenuations element pattern beamwidths azimuth elevation respectively definitions applied side replacing subscript fig shows one example antenna patterns gnbs ues assumed gnb arrays respectively size incumbent system assume fss antenna patterns least meet fcc regulation specified essentially regulation specifies minimum radiation suppression given angle centerline main beam fig shows normalized antenna gain given angle due high directivity antenna normalized antenna gain dbi angles degrees fig antenna pattern one dimension per fcc regulations table main parameters values applicable symbol disd ftbr max description height gnb distance number columns array number rows array antenna tilt antenna gain antenna transmit power beamwidth patterns azimuth beamwidth patterns elevation ratio loss noise figure channel bandwidth carrier frequency distance path loss shadowing standard deviation indicator variable denotes blockage event maximum antenna gain dbi value applicable disd ftbr ftbr ftbr ghz blockage blockage shown slight misalignment main boresight enough incur significant signal attenuation summary main parameters used provided table iii nalysis eployment section study deployment fss get guidelines deployment geometry features insights help understand deployment fss affects coexistence systems equally important also used benchmark modeling fss using approaches parse databases fss deployed four major metropolitan areas chicago new york los angeles san fransisco database covers area radius table table current number links pairs database database chicago new york los angeles san francisco links pairs summarizes analyzed databases link defined communication two fss whereas pair defined link unique spatial coordinates fss thus pair could multiple links different channel spatial distribution first analyze spatial distribution fss fig shows density variations region radius center region city center willis tower chicago empire state building new york financial districts los angeles san fransisco evident fss distributed space specifically tend higher density near city centers become sparsely deployed suburban areas overall fss low density relative existing cellular networks fig shows average height deployment given density shown except san francisco average height generally increases denser areas compared lightly dense areas showing deployment height appears correlated average building heights areas coexistence perspective implies density fss urban areas worrisome stations tend deployed altitudes cell sites contrast fss likely deployed relatively low heights suburban areas yet density low regions fig fig show cumulative density function cdf probability density function pdf fss deployment height average median heights least respectively importantly fss deployed metropolitan areas note fifth percentile relative ground many fss actually deployed hills since sites expected deployed heights four six meters gnbs majority fss limiting interference fss vice versa chicago new york los angeles san francisco chicago new york los angeles san francisco average height density per radius chicago chicago new york los angeles san francisco cdf fss height height san francisco height height height los angeles histogram new york histogram histogram average height given density density per density variations region radius cdf histogram height pdf fss height fig fss spatial deployment antenna specifications another critical aspect fss deployment physical antenna orientation fig shows histogram antenna tilt verifying vast majority fss tilt angles pointing horizontally instance fss tilt angles within degrees fss high negative tilts point street level fss however typically deployed top buildings verified fig words correlation deployment height negative tilt thus although fss higher chance experience interference point ground signals typically experience larger path loss given height fss another key feature fss high antenna gain indeed shown fig antenna gain typically high gains necessary coverage millimeter wave frequencies troublesome pairs vicinity reason maximum beamwidth per fcc regulations less equal verified fig vast majority fss beamwidths tilt degrees los angeles tilt degrees san francisco max antenna gain dbi los angeles max antenna gain dbi height tilt degrees chicago beamwidth degrees los angeles beamwidth degrees san francisco max antenna gain dbi new york max antenna gain dbi san francisco tilt degrees histogram histogram tilt degrees san francisco tilt degrees average height given tilt new york histogram histogram histogram histogram tilt degrees los angeles tilt degrees chicago tilt histograms tilt degrees histogram histogram new york height histogram chicago height histogram histogram histogram new york height chicago beamwidth degrees beamwidth histograms beamwidth degrees pdf fss height fig fss antenna information coexistence perspective must tightly aligned cause tangible interference otherwise signals highly attenuated falling outside receiver beam fig comments incumbent modeling coexistence aforementioned analysis different incumbents databases helps provide several modeling guidelines incumbent fss instance using popular homogeneous poisson point process hppp model locations fss may practical region interest large fss tend distributed space addition due disparities height deployment mmwave deployment meaningful consider stochastic processes processes third dimension constant reflects mean height buildings given area antenna parameters observed majority fss similar characteristics thus suffices assume antenna gain beamwidth assume point horizontally elevation coexistence perspective deployment strategy fss favorable future deployment following reasons fss generally deployed whereas cell sites meters ground street level deployment hence well fss vast majority fss oriented horizontally directed deployments fss point street level typically high altitudes increasing path loss beamwidths fss help significantly attenuate interfering signals fall outside main lobe nalysis nterference section present framework compute aggregate interference system incumbent systems approach used applicable coexistence two wireless communication systems rely directional beams summarized fig focus system operating uplink mode study interference fss following reasons first ues typically positive tilt angles compared gnbs thus former likely interfere fss second mobility ues makes locations appear random gnbs deployment optimized ensure minimal interference fss interference seen victim aggregation ues vicinity transmitting respective gnbs aggregated interference depends mainly three components path loss attenuation due antenna pattern iii attenuation due antenna pattern describe one details next follows denote interfering victim receiver corresponding transmitter respectively addition denotes distance denotes dot product two vectors path loss user fixed station signals significantly attenuated blocked objects mmwave frequencies critical consider whether link los nlos path loss computations compute path loss victim interferer compute attenuation victim antenna azimuth elevation compute interferer effective eirp azimuth elevation compute aggregate interference interferers victim fig general framework computing aggregate interference interfering transmitter victim receiver rely directional data communications high frequencies end use path loss model expressed pllos plnlos pllos los path loss plnlos nlos path loss shadow fading binary variable indicates whether blocked building indicator function note pllos plnlos functions distance heights center frequency given essentially path loss model different path loss exponents depending distance also standard deviation shadow fading depends whether link los nlos model probability function depends distance used determine whether link los nlos work rely actual building layouts determine events define blockage event blocked building computed follows assuming represents ground first check whether line connects blocked building defined polygon model generalized include indoor losses penetration losses ues located indoors shown fig blockage event occurs hbl polygon intersect line check whether blocks line version polygon third dimension building height hbl specifically let distance building distance blockage event occurs hbl tan tan visualized fig attenuation due antenna pattern illustrated fig small misalignment received signal antenna boresight results significant attenuation thus critical accurately compute interfering signal antenna define line connecting interference axis let azimuth angle angle antenna boresight interference axis unit vector azimuth direction antenna boresight kxf unit vector antenna towards similarly let kxu elevation angle shown tan antenna tilt angles shown fig finally combined azimuth elevation attenuation victim receiver expressed max min ftbr max maximum antenna gain dbi ftbr ratio loss ftbr attenuation given angle corresponds antenna pattern matches fcc regulations fig radiated power eirp antenna actual directions directions two opposite panels defined unit vectors direction panels boresight assume directions random azimuth boresight first one distributed uniformly one pointing opposite direction first one one antenna panels active data communications let ustr denote unit vector denote unit vector azimuth direction panel active also let ubeam azimuth direction main lobe beam used corresponds beam maximum received power user beam association similarly define vustr vubeam elevation directions total radiated power direction victim expressed dbm beam max str min ftbr max maximum antenna gain ftbr ftbr loss azimuth angles computed beam ubeam str ustr elevation angles computed beam tan tan denotes angle vector angles illustrated fig interference axis interference axis respect respect fig azimuth elevation angles random directions also present random model azimuth elevation directions model require deployment gnbs hence ignores computational complexity simulating user beam association model assumes uses beam beam direction antenna main boresight ubeam ustr vustr end model azimuth direction uniform random variable whereas elevation direction modeled tan disd constant work consider angles used compute unit vectors needed azimuth elevation angles aggregate interference interference caused given dbm similarly inr expressed inrdb iagg iagg dbm noise power spectral density bandwidth noise figure passive nterference itigation echniques section propose several interference mitigation techniques protect incumbent fss focus two critical aspects first techniques passive require coordination fss second practical implement appeal mobile operators vendors mitigation technique propose switch sectors creating exclusion zones key idea beam directions typically reciprocal gnbs thus reciprocal directions point fss must discouraged using sector reciprocal direction pointing towards fss switched formally let ustr unit vector direction sector boresight reciprocal direction sector switched otherwise predetermined decision threshold relaxed sector exclusion criterion switch sectors aligned antenna orientation rather location criterion still reduce interference experienced fss slight misalignment antenna incurs significant signal attenuation formally sector switched ustr refer mitigation mitigation techniques demonstrated fig mitigation mitigation four decisions need made priori gnb making approach simple implement however may result tangible coverage holes affecting performance system end make exclusion zones finer scale decisions made basis instead specifically beam switched otherwise ues connect sector ues connect sector sector based fig illustration passive mitigation techniques unit vector direction predetermined beam decision threshold ubeam beam words sector could beams switched beams switched depending whether beam meets criterion also make decisions based orientation instead location ubeam exclusion zone shown fig spatial power control aforementioned techniques classified angular exclusion zones leading inevitably lower downlink coverage higher degradation zones used instead alternative switching beams sectors implement almost blank quiet angular zones power control used beams thus would still connect gnb beam reciprocal direction aligned receiver yet transmit power reduced beam essence emulates like approach angular domain water levels proportional beam alignment receiver power control formulated optimization problem optimizing variables maximum power levels assigned beam objective function captures potential interference beam receiver seek simple power control leave sophisticated algorithms future research direction end merely consider binary power control algorithm total uplink transmit power pul pup otherwise index gnb beam connects words beams classified regular beams quiet beams note natural extend approach sectors make respect orientation instead location implementation implementation angular exclusion zones straightforward indeed upon deployment gnbs given region mobile operator must identify fss vicinity using fcc database operator extract locations azimuth directions used compute necessary unit vectors operator switch sectors beams depending protection criterion used thus ues find reference signals sectors beams hence connect user beam association clearly operator may need update decisions databased changed switch back sectors incumbent license expiring typically happens scale implement spatial binary power control operator must tag beam indicator variable embed value reference signal sent beam done physical broadcast channel xpbch epbch thus synchronization decode master system information blocks mib sib identifying transmit power limit beam lincoln park chicago loop lower downtown manhatta fig simulation scenarios denotes denotes gnb denotes imulation esults study aggregate interference fss deployed lincoln park chicago loop lower manhattan shown fig channel model assumed follow model main simulation parameters listed table assume instantaneous load available slots consider center frequencies assume maximum radiated power without attenuation per fcc regulations consider ftbr noise power assume computed temperature finally location height maximum antenna gain antenna tilt noise figure extracted database subsequent results averaged spatial realizations validation system first verify deployment gnbs lead reliable coverage ues fig shows cdf ratio snr side beam association whereas fig shows main snr statistics overall shown deployment provides reliable coverage positive snr values operating slight snr degradation due higher path loss compared operating also show distribution used gnb beams gnb sectors beams fig shown overall azimuth gnb beam equally likely used similar observation regarding gnb sectors importantly elevation beams active suggests mobile operators implement couple elevation beams serve lincoln park mean snr median median mean snr median chicago loop snr lower manhattan snr cdf cdf snr chicago loop lincoln park snr cdf lower manhattan mean snr cdf snr statistics fig snr outdoor users reduces complexity user association codebook design similar conclusion drawn side elevation beams used well majority less thus ues likely nearby fss deployed high altitudes finally likely use azimuth beams near boresight higher gain compared beams distribution inr fig shows cdf inr different case studies also show reference inr threshold corresponds sinr degradation meeting fcc interference protection criterion following observations first using random model random azimuth elevation directions provides accurate results match well computing actual pointing directions presence gnbs follows deployment gnbs agnostic locations fss distribution used elevation directions fig similar pdf one used random model fig second cdfs show inr overall low majority fss experiencing inr levels well noise floor follows due high attenuation millimeter wave frequencies networks operate regime stark height difference deploying fss systems low likelihood ues aligned within beam also shown dense urban areas downtown manhattan lower inr due increased blockage resulted presence buildings finally inr slightly lower compared due higher path loss former lincoln park histogram lincoln park histogram histogram azimuth beam degrees lower manhattan histogram histogram histogram histogram histogram histogram sector index gnb sectors elevation beam angle degrees lower manhattan azimuth beam angle degrees lower manhattan pdf histogram histogram sector index lower manhattan elevation beam angle degrees chicago loop azimuth beam angle degrees lincoln park sector index chicag loop gnb elevation beams lincoln park elevation beam angle degrees elevation beam angle degrees lower manhattan gnb azimuth beams elevation beam angle degrees chicago loop azimuth beam degrees histogram histogram histogram histogram azimuth beam degrees lincoln park azimuth beam angle degrees elevation beam angle degrees elevation angle degrees azimuth beams elevation beams pdf elevation angle random model fig distribution used beams sectors lincoln park actual random actual random cdf cdf lower manhattan actual random actual random cdf chicago loop actual random actual random threshold ratio lincoln park threshold ratio chicago loop threshold ratio lower manhattan fig cdf inr fig shows pdf inr different case studies fig shows main inr statistics mean median percentile seen fss may experience high inr values protection threshold lincoln park whereas rest well protected motivates implementing proposed mitigation ratio ratio pdf ratio ratio ratio lincoln park pdf pdf pdf pdf pdf ratio chicago loop lower manhattan fig pdf inr inr inr inr inr inr lincoln park percentile percentile percentile median median median inr mean inr inr mean inr mean chicago loop lower manhattan fig inr statistics techniques improve inr protection fss simplifying coexistence fig provides additional insightful details regarding different components interference framework specifically fig illustrates distribution different antenna attenuations due patterns whereas fig illustrates distribution attenuations total gain side general azimuth attenuation significant elevation attenuation direction respect boresight appear point randomly azimuth whereas elevation pointing upwards also clear elevation attenuation lowest lincoln park fss deployed relatively lower altitudes compared two dense urban areas explaining inr highest lincoln park fig compare theoretical los probability model simulated one resulted using different building layouts los azimuth azimuth probability los cdf cdf gain elevation cdf cdf total cdf elevation cdf lincoln park chicago loop lower manhattan attenuation attenuation attenuation lincoln park chicago loop lower manhattan attenuation attenuation probability model lincoln park chicago loop lower manhattan gain distance fig distribution different attenuations side distribution different attenuations side comparison model actual layouts terms los probability probability expressed exp exp plos min overall model underestimates los probability larger distances lincoln park chicago loop case lower manhattan due dense deployment buildings remark expect los probability lower blockage due objects included foliage cars making fss even better protected impact mitigation focus particular lincoln park relatively high inr comparison fss interest deployed height wide vicinity making susceptible interference system consider operating maximum radiated power leads highest interference fig shows average inr presence mitigation following observations first using low protection thresholds result tangible reduction percentile inr follows ues tend point randomly space even main lobe aligned still chance high interference side lobes reason larger thresholds provide much better protection second shown protection reliable threshold inr threshold mitigation location orientation location orientation location orientation mitigation location orientation location orientation location orientation cdf cdf inr fig inr presence absence passive mitigation implies get low inr enough protect boresight signal attenuation due pattern may sufficient effective radiated power high third mitigation slightly outperforms mitigation particularly high thresholds equally important former also enables better coverage makes decisions higher angular resolution compared cost using mitigation increased number decisions needed made gnb vicinity fig shows main inr statistics variations angular protection threshold show inr performance absence mitigation reference also show inr presence spatial exclusion zones radii gnbs deployed inside zones expected inr significantly deceased high protection thresholds instance percentile decreases approximately locationbased beam mitigation used respectively angular exclusion zones effective spatial exclusions latter leads coverage holes system also emphasizes interference dominated ues close rather ues beams directed towards boresight fig shows one snapshot interest system vicinity without mitigation techniques snapshot show beam used data communication associated gnb well interference generated dbm fig inr high dominated interference noise floor approximately using mean median inr threshold degrees mitigation exc zone exc zone percentile inr inr threshold degrees threshold degrees fig inr statistics variations angular protection threshold sector mitigation shown fig particular switches different gnb reducing interference similar observation made beambased approach illustrated fig use showing angular exclusion zones finer scale sufficient protect without compromising coverage impact spatial power control set plo pup maximum radiated power respectively fig shows inr cdf power control used evident higher protection thresholds power control effective reduce inr instance percentile reduces finally fig shows main inr statistics variations protection threshold shown percentile reduced approximately without need shut beams comparison mitigation techniques aforementioned techniques shown effectiveness mitigating interference section compare terms impact coverage system using gnb antenna parameters shown maximum radiated power fig shows comparison different techniques terms coverage consider protection due angular exclusion zones created using larger thresholds inevitably affect coverage case spatial power control beams sectors active fig shows curves different mitigation techniques curves highlight different possible operating points coexisting incumbent systems interference level expected incumbent target coverage following mitigation fig one simulation snapshot experiences high inr absence mitigation beams used ues shown interference generated one given dbm mitigation using exclusion zone using exclusion zone observations comparing beam angular zones sector angular zones evident former presents operating points making flexible effective however inevitable higher inr protection incurs coverage degradation addition spatial exclusion zones effective angular exclusion zones instance using approximation simulated curves shown slopes cdf mitigation inr fig inr cdf presence absence power control mean median mitigation zone exc zone inr inr inr percentile threshold degrees threshold degrees threshold degrees fig inr statistics spatial power control used mean threshold degrees snr median snr snr mitigation exc zone threshold degrees threshold degrees fig inr statistics spatial power control used spatial exclusion zone techniques approximately words reducing median snr inr reduces using angular exclusion zones using spatial exclusion zones finally spatial power control enables reduction inr negligible coverage loss vii onclusion spectrum bands potential enable true mobile connectivity gigabit speeds key obstacle deployment bands presence incumbent systems require protection harmful interference end percentile inr mitigation exclusion zone mitigation exclusion zone percentile inr median snr snr median snr percentile inr snr percentile inr fig comparison curves different mitigation techniques thoroughly analyzed existing databases understand key features properties incumbent system including spatial distribution antenna specifications addition analyzed aggregate interference ues using realistic channel models actual building layouts accurate results analysis results revealed coexistence beyond feasible thanks high propagation losses millimeter wave frequencies high attenuation due misalignment antenna boresight deployment geometry fss tend systems fss deployed relatively low altitudes proposed several passive mitigation techniques including angular exclusion zones spatial power control techniques require minimal effort mobile operators require coordination incumbents shown exclusion zones angular domain effective spatial exclusion zones however coverage degrade due coverage holes introduced exclusion zones overcome via power control transmit power depends beam direction respect victim eferences hattab visotsky coexistence mmwave users incumbent fixed stations ghz ieee globecom xiao mumtaz millimeter wave communications future mobile networks ieee sel areas vol imt framework overall objectives future development imt beyond andrews buzzi ieee sel areas vol jun use spectrum bands ghz mobile radio services federal communications commission fcc tech docket july itu world radiocommunication conference geneva switzerland rangan rappaport cellular wireless networks potentials challenges proceedings ieee vol mar cudak kovarik experimental wave cellular system proc ieee globecom workshops inoue yoshioka field experimental trials mobile communication system using proc ieee wireless communications networking conf workshops wcncw mar huawei technologies huawei bring mmwave live demo deutsche telekom online available http gupta alkhateeb gains restricted secondary licensing millimeter wave cellular systems ieee journal selected areas communications vol rebato boccardi hybrid spectrum sharing mmwave cellular networks ieee transactions cognitive communications networking vol jun boccardi spectrum pooling mmwave networks opportunities challenges enablers ieee communications magazine vol ghorbanzadeh visotsky radar inband interference lte macro small cell uplinks ghz band proc ieee wireless communications networking conf wcnc mar radar interference lte base stations ghz band physical communication vol online available http khawar abdelhadi coexistence analysis radar cellular system los channel ieee antennas wireless propag vol beltran ray understanding current operation future roles wireless networks competition unlicensed spectrum bands ieee journal selected areas communications vol kim xian study coexistence systems systems fixed service ghz band proc ieee int microwave symp may numerical simulation study frequency sharing systems fixed service systems bands ieee access vol kim visotsky coexistence incumbents ghz bands ieee sel areas vol jun alkhateeb nam initial beam association millimeter wave cellular systems analysis design insights ieee trans wireless vol may giordani mezzavilla initial access mmwave cellular networks ieee communications magazine vol barati hosseini initial access millimeter wave cellular systems ieee transactions wireless communications vol skolnik introduction radar systems vol study channel model frequency spectrum generation partnership project tech title code federal regulations federal communications commission fcc tech rep section july haenggi stochastic geometry wireless networks cambridge university press
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quantum ensembles quantum classifiers maria schuld quantum research group school chemistry physics university durban south africa apr schuld francesco quantum research group school chemistry physics university durban south africa national institute theoretical physics petruccione abstract quantum machine learning witnesses increasing amount quantum algorithms decision making problem potential applications ranging automated image recognition medical diagnosis many algorithms implementations quantum classifiers models classification data inputs quantum computer following success collective decision making ensembles classical machine learning paper introduces concept quantum ensembles quantum classifiers creating ensemble corresponds state preparation routine quantum classifiers evaluated parallel combined decision accessed measurement framework naturally allows exponentially large ensembles similar bayesian learning individual classifiers trained example analyse exponentially large quantum ensemble classifier weighed according performance classifying training data leading new results quantum well classical machine learning pacs numbers keywords quantum machine learning quantum algorithms ensemble methods committee decisions introduction machine learning classifier understood mathematical model computer algorithm takes input vectors features assigns classes labels example features could derived image label assigns image classes shows cat shows cat classifier written function mapping input space space possible labels quantum ensembles quantum classifiers function depend set parameters training classifier refers fitting parameters sample data pairs order get model generalises data classify future inputs common practise consult one trained model train ensemble models derive collective prediction supersedes predictive power single classifier emerging discipline quantum machine learning number quantum algorithms classification proposed demonstrate train use models classification quantum computer open question cast quantum classifiers ensemble framework likewise harvests strengths quantum computing article first step answering question focus special type quantum classifier namely model set parameters expressed state classification result encoded separate register qubits format allows create superposition quantum models evaluate parallel state preparation scheme used weigh classifier thereby creating ensemble allows instantaneous evaluation exponentially large quantum ensembles quantum classifiers exponentially large ensembles potential increase predictive power single quantum classifiers also offer interesting perspective circumvent training problem quantum machine learning training quantum regime relies methods range sampling quantum states quantum matrix inversion grover search however complex optimisation problems little mathematical structure given important example neural networks translation iterative methods backpropagation efficient quantum algorithms less straight forward see ref known classical machine learning one avoid optimisation exchanging integration bayesian learning one solve integral possible parameters instead searching single optimal candidate idea integrating parameters understood forming collective decision consulting possible models certain family weigh according desired influence approach ensemble framework words quantum ensembles quantum classifiers offer interesting perspective learning quantum computers order illustrate concept quantum ensembles quantum classifiers investigate exponentially large ensemble inspired bayesian learning every ensemble member weighed according accuracy data set give quantum circuit prepare quantum ensemble general quantum classifiers collective decision computed parallel evaluated single qubit measurement turns certain models procedure quantum ensembles quantum classifiers tively constructs ensemble accurate classifiers perform training set better random guessing shown cases accurate weak classifiers build strong classifier analytical numerical investigations show may work knowledge result established classical machine learning literature shows quantum machine learning stimulate new approaches traditional machine learning well article structured follows section provides definitions summarises research relevant work section introuduces formalism quantum ensembles quantum classifiers section presents example accuracyweighted ensemble shows cases effective ensemble accurate members section analyses example analytically leads conclusion section classification asymptotically large ensembles accurate models introducing quantum ensemble framework well example ensemble provide review classical results ensemble methods emphasis lies studies analyse asymptotically large ensembles accurate classifiers although rigorous proofs available find strong arguments high predictive power large collections weak learners problem focus supervised binary pattern classification task given dataset inputs outputs labels well new input goal predict unknown label consider classifier input parameters mentioned common approach machine learning choose model fitting parameters data ensemble methods based notion allowing one final model prediction whatever intricate training procedure neglect strengths candidates even overall worse performance example one model might learned deal outliers well expense slightly worse predicting rest inputs expert knowledge lost one winner selected idea allow ensemble committee trained models sometimes called experts members take decision new prediction together considering familiar principle societies surprising thought gained widespread attention late many different proposals put forward use one model prediction proposals categorised along two dimensions first selection procedure apply obtain ensemble members second quantum ensembles quantum classifiers decision procedure defined compute final output see figure note discuss ensembles built different machine learning methods consider parametrised model family given fixed hyperparameters example neural network fixed architecture adjustable weight parameters strategy constructing ensemble train several models decide according majority rule intricate variations popular practice interesting theoretical foundations bagging trains classifiers different subsamples training set thereby reducing variance prediction adaboost trains subsequent models part training set misclassified previously applies given weighing scheme understood reduce bias prediction mixtures experts train number classifiers using specific error function second step train gating network defines weighing scheme methods ensemble classifier written sgn coefficients weigh decision model ensemble specified sign function assigns class new input weighed sum positive otherwise important following rewrite sum possible parameters use finite number representation limit parameters certain interval get discrete sum sgn continuous limit sum replaced integral order obtain ensemble classifier weights correspond models part ensemble set zero given model family interval parameters well precision represented ensemble therefore fully defined set weights writing sum possible models provides framework think asymptotically large ensembles realised quantum parallelism interesting enough formulation also close another paradigm classification bayesian learning approach given training dataset understanding well random variables goal classification bayesian setting find conditional probability distribution prediction derived example maximum posteriori estimate first part integrand probability pair observed given set parameters chosen model correspondence becomes apparent one associates quantum ensembles quantum classifiers selection procedure decision procedure figure principle ensemble methods select set classifiers combine predictions obtain better performance generalising data classifiers considered parametrised functions family set parameters solely defines individual model dataset consulted selection procedure sometimes also plays role decision procedure label new input chosen interprets estimator posterior true model given observed data also consider different model families specified hyperparameters method turns bayesian model averaging sometimes included canon ensemble method although based rather different theoretical foundation beyond transition bayesian framework increasing size ensemble include possible parameters studied different contexts cases adding accurate classifiers shown increase performance ensemble decision accurate classifiers accuracy estimated number correctly classified test samples divided total samples validation set thus better random guessing means learned pattern training set least small extend case developed schapire leading aforementioned adaboost algorithm collection weak classifiers accuracy slightly better turned strong classifier expected high predictive power advantage weak classifiers comparably easy train combine people thought power weak learners long adaboost cordocet jury theorem states considering committee judges judge probability reach correct decision probability correct collective decision majority vote converge number judges approaches infinity idea applied ensembles neural networks hansen salamon ensemble members likelihood classify new instance correctly errors uncorrelated probability majority rule classifies new instance incorrectly given quantum ensembles quantum classifiers error ensemble size figure left prediction error increasing size ensemble classifiers accuracy asymptotically error converges zero right odds ratio grows slower square together results lam described text indication adding accurate classifiers ensemble high chance increase predictive power ensemble size convergence behaviour plotted figure left different values assumption uncorrelated errors idealistic since data points difficult classify others therefore tend misclassified large proportion ensemble members hansen salamon argue highly overparametrised neural network models consider base classifiers training get stuck different local minima ensemble members sufficiently diverse errors realistic setting would also assume model different prediction probability measure accuracy investigated lam suen change prediction power growth ensemble obviously depends predictive power new ensemble member sign determined roughly stated adding two classifiers accuracies ensemble size increase prediction power value less odds ratio ensemble member plotting odds ratio square figure right becomes apparent chances high increase predictive power ensemble adding new weak learner together results literature results suggest constructing large ensembles accurate classifiers lead strong combined classifier proceeding quantum models another result important mention consider possible parameters sum equation assume model defined accuracy training set optimal weighing scheme given log interesting note weighing scheme corresponds weights chosen adaboost trained model derived seems different theoretical objective quantum ensembles quantum classifiers figure example representation parameters uniform quantum superposition three qubits computational basis state corresponds parameter interval quantum ensembles quantum classifiers idea section cast notion ensembles quantum algorithmic framework based idea quantum parallelism used evaluate ensemble members exponentially large ensemble one step consider quantum routine computes model function call quantum classifier following last qubit thereby encodes class state class state note important whether registers encode classical vectors amplitudes qubits quantum state encoding classical information binary sequence computational basis states every function classical computer compute efficiently could principle translated quantum circuit means every classifier leads efficient quantum classifier possibly large polynomial overhead example quantum perceptron classifier found ref neural networks considered definition quantum classifier implemented parallel superposition parameter states example given expression could uniform superposition computational basis state corresponds length binary representations parameter dividing interval limits parameter values candidates see figure explained ensemble method understood weighing rule model sum therefore require second quantum routine turns uniform superposition one quantum ensembles quantum classifiers weighing model classical probability call routine quantum ensemble since weights define model contributes extend comp note bined decision normalisation factor ensures routine understood state preparation protocol qsample together quantum routines define quantum ensemble quantum classifiers first weighing superposition models words basis states parameter register computing model predictions new input parallel final quantum state given measurement statistics last qubit contain ensemble prediction chance measuring qubit state probability ensemble deciding class given chance measuring qubit state reveals given subset containing models describing general template look implement specific weighing scheme quantum routine general models analyse resulting classifier detail choosing weights proportional accuracy illustrative case study choose weights proportional accuracy model measured training set precisely proportion correct classifications number data points note usually estimate accuracy trained model measured separate test validation set require accuracy build classifier first place therefore measure training set goal quantum algorithm prepare quantum state model represented state probability measured proportional accuracy quantum ensembles quantum classifiers weighing routine constructed follows required computation system qubits divided four registers data register parameter register output register accuracy register qubits qubits assume quantum classification routine given first step hadamards bring parameter register uniform superposition hadamard applied accuracy qubit thereby encodes set parameters load training pairs successively data register compute outputs superposition applying core routine rotate accuracy qubit small amount towards depending whether target output actual output value xor gate respective qubits together conditional rotation last qubit core routine loading step uncomputed applying inverse operations training points processed accuracy qubit entangled parameter register state postselective measurement accuracy qubit accepts state repeats routine otherwise selects superposition leaves state normalisation factor equal acceptance probability pacc probability acceptance influences runtime algorithm since measurement ancilla means abandon result start routine scratch expect choices parameter intervals data allows keep acceptance probability sufficiently high many machine learning applications expected distributed around hints towards rejection sampling promising tool translate quantum ensemble classical method load new input first qubits data register apply routine uncompute disregard data register obtain one could alternatively prepare training superposition trace training register end costs remain linear number training vectors times training vector strategies quantum ensembles quantum classifiers class class new input figure illustration decision boundary perceptron model twodimensional inputs parameter vector defines hyperplane dividing input space regions class change sign parameters swaps decision regions leads accuracy measurement statistics last qubit contain desired value precisely expectation value given corresponds classifier repeated measurements reveal expectation value desired precision accuracies may good weights turn question accuracies might good weighing scheme recall lot evidence ensembles weak accurate classifiers meaning lead strong classifier ensemble constructed last section however contains sorts models undergo selection procedure may therefore contain large even share models low accuracy random guessing turns large class model families ensemble effectively contains accurate models knowledge new result also interesting classical ensemble methods assume core machine learning model property point symmetric parameters true linear models neural networks odd number layers simple perceptron architecture hidden layers let furthermore assume parameter space symmetric around zero meaning quantum ensembles quantum classifiers weight log accuracy figure comparison effective weight transformation described text optimal weight rule flip prediction classifier accuracy less also pairs denoted notation one write follows get intuition consider linear model imposing linear decision boundary input space parameters define vector orthogonal decision boundary addition bias term defines boundary intersects ignore moment sign change parameters flips vector around linear decision boundary remains exactly position meanwhile decisions turned around see figure point symmetric models expectation expressed sum one half parameter space result effective transformation shift weights interval profound consequences transformation plotted figure one see accurate models get positive weight models get negative weight random guessers vanish sum negative weight consequently flips decision bad models turns accurate classifiers linearisation rule mentioned optimal weight distribution large ensembles plotted black comparison mind rewrite expectation value quantum ensembles quantum classifiers new weights parameters well one assume parameter space still wants construct ensemble accurate models alternative sketched routine could construct oracle marks models use amplitude amplification create desired distribution way mark models load every training vector predefined register compute perform binary addition separate register counts number correctly classified training inputs register would binary encoding count hence require dlog qubits well garbage qubits addition operation log integer first qubit would one number correct classifications larger zero otherwise words qubit would flag accurate models used oracle amplitude amplification optimal number grover iterations depends number flagged models best case around models accurate optimal number iterations order course number estimated performing grover analytical investigation ensemble order explore ensemble classifier conduct analytical numerical investigations remainder article convenient assume know probability distribution data picked either true probability distribution data generated approximate distribution inferred data mining technique furthermore consider continuous limit parameter defines decision regions input space class class regions inputs mapped respective classes accuracy expressed words expression measures much density class falls decision region proposed model class good parameters propose decision regions contain areas class distribution factor necessary ensure accuracy always interval since one analytically prove possible models accurate case pointsymmetric functions one artificially extend superposition twice size prevent half subspace flagged thereby achieve optimal amplitude amplification scheme quantum ensembles quantum classifiers class class class class figure simple model classifier analysed text parameter denotes position decision boundary parameter determines orientation two probability densities normalised probability distributions consider form normalised vanish asymptotically hyperparameters define mean position variance width distribution two classes prominent examples gaussians box functions let integral fulfils two expressions following property become important later consider minimal toy example classifier namely perceptron model input space sgn one parameter would sufficient mark position decision boundary second one required define orientation one simplify model even letting bias define position decision boundary introducing binary orientation parameter illustrated figure sgn simplified perceptron model decision regions given orientation goal compute expectation value quantum ensembles quantum classifiers figure black line plots expectation value ensemble decision discussed text two class densities different new inputs positive expectation value yields prediction negative expectation value predicts class lies decision boundary plot shows model predicts decision boundary would expect namely exactly two distributions class class figure perceptron classifier inputs bias ensemble consist models three parameters taking equally distributed values resolution compute decision regions dataset generated python blob function classes variance parameter one see two dimensions decision boundary still lies two means sign function evaluates desired prediction inserting definitions well calculus using properties brings expression sgn evaluating sign function two cases leads analyse expression consider two class densities gaussian probability distributions exp quantum ensembles quantum classifiers indefinite integral gaussian distributions given erf inserting get expectation value erf erf figure plots expectation value different inputs case two gaussians decision boundary point expectation value changes negative positive value one see simple case decision boundary two means would naturally expect important finding since implies ensemble classifier works arguably simple model dataset quick calculation shows always find decision boundary midway two means case integrals evaluate lim erf erf lim integral error function erf assuming mean variance two class distributions reasonable finite value error function evaluates limit process becomes important one therefore write lim lim lim lim lim expectation value case equal variances therefore becomes setting turns expectation value zero point two variances shown decision boundary simulations confirm also true distributions square exponential lorentz distribution well data see figure quantum ensembles quantum classifiers example prediction class distributions accuracy resulting function example prediction class distributions accuracy resulting function figure detailed analysis classification two examples normal data distributions upper example shows lower example shows plotted top left figure four top right figure block shows classification given new input varying parameters plotted respectively bottom left shows accuracy classification performance data distribution bottom right plots product previous two well resulting function integral prediction outcome integral black curve total gray shaded areas one see models different variances accuracies loose symmetry decision boundary therefore lie middle two means simplicity core model allows look structure expectation value figure shows components integrand quantum ensembles quantum classifiers expectation value namely accuracy core model function well product example shows variances example plots different variances plots show equal variances accuracy symmetric function centred two means different variances function highly asymmetric case two distributions sufficiently close sensitive impact position decision boundary shifted towards flatter distribution might desired behaviour contexts first foremost interesting result analytical investigation summary analysing simple classifier relation gaussian data distributions gives evidence weighted average ensemble classifier lead meaningful predictions separated data seems sensitive dependency shape distributions investigations would show ensemble classifier behaves complex base classifiers realistic datasets low resolution simulations inputs confirm nonlinear decision boundaries fact learnt however computing exact expectation value huge computational effort example next complex neural network model requires two hidden layers point symmetric one bias neuron inputs architecture already seven weights weight encoded three qubits including one qubit sign get ensemble members whose collective decision needs computed entire input space order determine decision boundaries sampling methods could help obtain approximations result would open methods classical applications well conclusion article proposed framework construct quantum ensembles quantum classifiers use parallelism order evaluate predictions exponentially large ensembles proposal leaves lot space research first mentioned quantum ensemble interesting extensions classical methods considering approximations compute weighted sum members predictions recent results quantum supremacy boson sampling show distributions state preparation routines therefore quantum ensembles computed efficiently classical computers meaningful quantum ensembles speedup proven rules combining weak learners strong learner different quantum regime types ensembles generically prepared quantum devices second important issue touch upon overfitting adaboost regularisation equivalent early stopping bayesian learning inbuilt regularisation mechanisms ensemble relate cases likely overfit considering flexible models third question whether coherent quantum ensembles quantum classifiers ways defining quantum ensemble quantum machine learning algorithms format quantum classifier defined mixed states used context summary article provided first step think ensemble methods quantum machine learning example mutual enrichment classical quantum machine learning provide acknowledgments work supported south african research chair initiative department science technology national research foundation thank ludmila kuncheva thomas dietterich helpful comments references thomas dietterich ensemble methods machine learning international workshop multiple classifier systems pages springer patrick rebentrost masoud mohseni seth lloyd quantum support vector machine big data classification physcial review letters sep seth lloyd masoud mohseni patrick rebentrost quantum algorithms supervised unsupervised machine learning arxiv preprint mohammad amin evgeny andriyash jason rolfe bohdan kulchytskyy roger melko quantum boltzmann machine arxiv preprint ashish kapoor nathan wiebe krysta svore quantum perceptron models advances neural information processing systems pages patrick rebentrost maria schuld francesco petruccione seth lloyd quantum gradient descent newton method constrained polynomial optimization arxiv preprint ludmila kuncheva combining pattern classifiers methods algorithms john wiley sons leo breiman random forests machine learning robert schapire strength weak learnability machine learning yoav freund robert schapire generalization learning application boosting european conference computational learning theory pages springer robert jacobs michael jordan steven nowlan geoffrey hinton adaptive mixtures local experts neural computation zoubin ghahramani probabilistic machine learning artificial intelligence nature richard duda peter hart david stork pattern classification john wiley sons new york thomas minka bayesian model averaging model combination comment available electronically http html lars kai hansen peter salamon neural network ensembles ieee transactions pattern analysis machine intelligence louisa lam suen application majority voting pattern recognition analysis behavior performance ieee transactions systems man systems humans quantum ensembles quantum classifiers lloyd shapley bernard grofman optimizing group judgmental accuracy presence interdependencies public choice maria schuld ilya sinayskiy francesco petruccione simulate perceptron using quantum circuits physics letters volume pages kwok wan oscar dahlsten robert gardner kim quantum generalisation feedforward neural networks arxiv preprint robert schapire explaining adaboost empirical inference pages springer
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learning deep object detectors models xingchao peng baochen sun karim ali kate saenko university massachusetts lowell oct xpeng bsun karim saenko abstract crowdsourced cad models becoming easily accessible online potentially generate infinite number training images almost object category show augmenting training data contemporary deep convolutional neural net dcnn models synthetic data effective especially real training data limited well matched target domain freely available cad models capture shape often missing low level cues realistic object texture pose background detailed analysis use synthetic images probe ability dcnn learn without cues surprising findings particular show dcnn target detection task exhibits large degree invariance missing cues pretrained generic imagenet classification learns better cues simulated show synthetic dcnn training approach significantly outperforms previous methods pascal dataset learning scenario improves performance domain shift scenario office benchmark figure propose train object detectors real images augmenting training data synthetic images generated freely available cad models objects collected used successfully past add affine transformations training images recognize text even train detectors handful categories cars however yet demonstrated detection many categories modern dcnns trained object detectors categories synthetic cad images used hog gradient model dpm significantly less powerful dcnns object classification detection main challenge training freely available cad models capture shape object frequently lack cues object texture background realistic pose lighting etc used simple rendering objects uniform gray texture white background showed models learn well data invariant color texture mostly retain overall shape object however dcnn visualizations shown retain color texture patterns therefore unknown would tolerate lack cues training images sophisticated rendering process simulates cues needed investigate missing cues affect dcnns ability learn object detectors study precise nature cue invariances given object category dcnn maps cues contained image shape texture category information cat car represented top layer activations alexnet define cue invariance ability network extract equivalent category introduction deep cnn models achieve performance object detection heavily dependent largescale training data unfortunately labeling images detection extremely every instance every object must marked bounding box even largest challenge datasets provide limited number annotated categories categories pascal voc coco imagenet wanted train detector novel category may feasible compile annotate extensive training set covering possible variations propose bypass expensive collection annotation real images using freely available cad models automatically generate synthetic training images see figure synthetic data augmentation figure learn deep detectors real images cad models explore invariance deep features missing cues shape pose texture context propose improved method learning synthetic cad data simulates cues information despite missing cues expect network learn different invariances depending task trained quantifying invariances could help better understand dcnn models impove transfer new domains data small number papers started looking problem many open questions remain dcnns invariant object color texture context pose invariance transferable new tasks help images synthetically rendered models design series experiments peer depths dcnns analyse invariance cues including ones difficult isolate using real image data make surprising discoveries regarding representational power deep features particular show encode far complex invariances cues pose color texture context previously accounted also quantify degree learned invariances specific training task based analysis propose method zeroor learning novel object categories generates synthetic data using models texture scene images related category advantage approach drastically reduces amount human supervision traditional labeling methods could greatly expand available sources visual knowledge allow learning detectors millions cad models available web present experiments pascal voc detection task show training data missing limited novel category method outperforms training real data synthetic method also demonstrate advantage approach setting real training data comes different domain target data using office benchmark summarize contributions gain new important insights cue invariance dcnns use synthetic data show synthetic training modern dcnns improves detection performance fewshot scenarios present evaluation synthetic cad training object detectors date related work object detection flat representations hog sift etc dominated object detection literature due considerable invariance factors illumination contrast small translations combination discriminative classifiers linear svm latent svm proved powerful learning localize global outline object recently convolutional neural networks overtaken flat features clear many image understanding tasks including object detection dcnns learn layered features starting familiar pooled edges first layer progressing complex patterns increasing spatial support extensions detection included cnn rcnn understanding deep cnns increasing interest understanding information encoded highly nonlinear deep layers reversed computation find image patches highly activate isolated neuron detailed study happens one transfers network layers one dataset another presented reconstruct image one layer activations using image priors recover natural statistics removed network filters visualizations confirm progressively invariant abstract representation image formed successive layers analyse nature invariances invariance simple transformations reflection rotation explored paper study complex invariances deconstructing image shape texture factors seeing specific combinations result representations discriminant object categories use synthetic data use synthetic data longstanding history computer vision among earliest attempts used models primary source information build object models recently used cad models source labeled data limited work categories like cars motorcycles utilized synthetic data probe invariances features like sift slf etc paper generate training data crowdsourced cad models noisy free available many categories evaluate approach categories pascal dataset much larger realistic previous benchmarks previous works designed special features matching synthetic object models real image data used hog features linear svms employ powerful deep convolutional images features demonstrate advantage directy comparing authors use cad models show results detection pose estimation train detectors real images labeled pose avoid expensive manual bounding box pose annotation show results minimum real image labels finally several approaches used synthetic training data tasks object detection example recently proposed synthetic text generation engine perform text recognition natural scenes proposed technique improve synthesis images using structural information models approach approach learns detectors objects training examples augmenting training data synthetic cad images overview approach shown figure given set cad models object generates synthetic image training dataset simulating various cues section extracts positive negative patches object synthetic images optional small number real images patch fed deep neural network computes feature activations used train final classifier deep detection method rcnn section explore cue invariance networks trained different ways described section synthetic generation cues realistic object appearance depends many cues including object shape pose surface color reflectance location spectral distributions illumination sources properties background scene camera characteristics others choose subset factors easily modeled using computer graphics techniques namely object texture color pose shape well background scene texture color learning detection model new category limited labeled real data choice whether simulate cues synthetic data depends invariance representation example representation invariant color grayscale images rendered study invariance dcnn representation parameters using synthetic data generated follows models viewpoints crowdsourced cad models thousands objects becoming freely available online start downloading models warehouse searching name desired object categories category around models obtained experiments explore effect varying shape restricting number models experiments original poses cad models arbitrary chairs tilted cars therefore adjust cad models viewpoint manually views shown figure best represent intraclass pose variance real objects next manually specified model view generate several small perturbations adding random rotation finally pose perturbation select texture color background image render virtual image include virtual training dataset next describe detailed process factors color texture investigate various combinations color texture cues object background image previous work shown learning detectors virtual data using hog features rendering natural backgrounds texture helpful equally good results obtained white background uniform gray object texture explain fact classifier focused learning outlines object shape invariant color texture hypothesise case different dcnn representations neurons shown respond detailed textures colors patterns explore invariance dcnns factors specifially examine invariance dcnn representation two types object textures realistic color textures uniform grayscale textures texture case background scenes examine invariance three types scenes namely color scenes grayscale scenes plain white background examples texture background generation settings shown table order simulate realistic object textures use small number per category real images containing real objects extract textures therein annotating bounding box texture images stretched fit cad models likewise order simulate realistic background scenes gathered per category real images scenes category likely appear blue sky images aeroplane images lake ocean boat etc generating virtual image first randomly select background image available background pool project onto image plane select random texture image texture pool map onto cad model rendering object obtain deep feature representation images use alexnet architecture million parameters network first achieved breakthrough results image classification remains studied widely used visual convnet network trained fully supervised backpropagation takes raw rgb image pixels fixed size outputs object category labels layer consists set neurons linear weights input followed nonlinearity first five layers network local spatial support convolutional final three layers fullyconnected neuron previous layer thus include inputs entire image network originally designed classification applied detection rcnn impressive gains popular object detection benchmarks adapt alexnet detection rcnn applied network image proposed selective search method adding background label applied suppression outputs hidden layers resulted performance improvements refer reader details alexnet feature representation since learned cue invariance example consider cat object class network invariant cat texture produce similar activations cats without texture hallucinate right texture given texureless cat shape detector learn cats equally well sets training data hand network invariant cat texture feature distributions differ classifier trained textureless cat data perform worse expect network learn different cue invariances depending task categories trained example may choose focus texture cue detecting leopards shape context texture unique evaluate effect compare three different variants network one generic imagenet ilsvrc classification task imgnet network additionally pascal detection task case category labels imgnet network synthetic cad data vcnn obtain vcnn network entire network synthetic data backpropagating gradients lower learning rate effect adapting hidden layer parameters synthetic data also allows network learn new information object categories synthetic data thus gain new objectclass invariances show essential good performance scenario treating network activations fixed features inferior learning capacity hidden layers final classifier investigate degree presence different cues affects well network learn synthetic data analysing cue invariance dcnn features experiments recall define cue invariance ability network extract category information training images despite missing cues object texture test invariance create two synthetic training sets one one without particular cue extract deep features sets train two object detectors compare performance real test data hypothesis representation invariant cue similar neurons activate whether cue present input image leading similar information training thus similar performance hand features invariant missing cue result missing category information poorer performance work extract last hidden layer cue invariance results deep convolutional neural network features first evaluate variations cues affect features generated imgnet networks pascal dataset experiment follow steps see figure select cues generate batch synthetic images cues sample positive negative patches class extract hidden dcnn layer activations patches features train classifier object category test classifiers real pascal images report mean average precision map determine optimal number synthetic training images computed map function size training set using image generation setting table results shown figure figure relationship map number training images generation setting dicate classifier achieves peak performance around training images positive instances categories number used subsequent experiments object color texture context experiment used pose perturbations per view views per category trained series detectors several background object texture cue configurations results shown table first expected see training synthetic data obtains lower mean training real data around bounding box regression also imgnet network representation achieves lower performance network case real data however somewhat unexpected result generation settings rgrr achieve comparable performance despite fact texture context results real texture color background rgrr best thus network learned invariant color texture object background also note settings achieve much lower performance points lower potentially uniform object texture well distinguished backgrounds imgnet network trend similar best performing methods means adding realistic context texture statistics helps classifier thus imgnet network less invariant factors least categories dataset note imgnet network seen categories training part ilsvrc classification task explains still fairly insensitive combinations uniform texture real background also perform well interestingly well networks leading conclusion networks learned associate right context colors objects also see variations across categories categories like cat sheep benefit adding object texture cue explore lower layers invariance color texture background visualize patches strongest activations units shown figure value receptive field corner normalized dividing max activation value units channel results interesting unit left subfigure fires patches resembling real images using synthetic data unit still fires even though background texture removed unit right fires white animals green backgrounds real images continues fire synthetic sheep simulated texture despite lack green background however fails images demonstrating specificity object color texture synthetic pose also analyse invariance cnn features object pose successive operations convolution cnns invariance translations scale likewise visualizations learned filters early layers indicate invariance local rotations thus cnn representation invariant slight translation rotations deformations remains unclear extent cnn representation large rotations experiment fix cad models three dominant poses shown table change number views used experiment keep total number synthetic training images exactly generating random small perturbations degree around main view results indicate networks adding side view front view gives boost improvement adding third view marginal note adding views may even hurt performance pascal test set may objects views real image pose also test view invariance real images interested objects whose frontal view presentation differs significantly horse frontal view end selected categories pascal voc training set match criteria held categories included rotationally invariant objects bottles tables next split training data categories prominent shown table train classifiers exclusively removing one view say test resulting detector pascal voc test set containing side also compare random view sampling results shown table point important surprising conclusions regarding representational power cnn features note map drops less detectors exclusively trained removing either view tested pascal voc test set detectors never presented second view also trained approximately half data invariance large complex pose changes may explained fact cnn model trained views object present subsequently views real rgb real rgb aero bike imgnet aero bike bird bird white real rgb boat botl boat botl bus bus white unif gray car cat chr car cat chr cow cow real rgb unif gray tab dog hse tab dog hse real gray unif gray mbik pers plt shp mbik pers plt shp sofa sofa real gray real rgb trn map trn map table detection results pascal test dataset row trained different background texture configuration virtual data shown top table middle table dcnn trained imagenet ilsvrc classification data finetuned pascal training data bottom table network pascal figure top regions strongest activations units using method overlay unit receptive field drawn white normalized activation value shown corner unit show results top bottom real pascal images see text explanation present level invariance nevertheless remarkable last experiment reduce training set removing objects note larger map drop points much less one may expect conclude networks representation groups together multiple views object shape finally experiment reducing intraclass shape variation using fewer cad models per category otherwise use settings condition experiments find map decreases points using half models shows significant boost adding shape variation training data indicating less invariance factor learning results pascal summarize conclusions previous section found dcnns learn significant amount invariance texture color pose less invariance shape trained task trained task degree invariance lower therefore learning detection model new category limited labeled real data available advantageous simulate factors synthetic data section experiment adapting deep representation synthetic data use available models views compare two generation settings produced best results table settings use realistic backgrounds may advantages detection particular visualizations positive training data show white background around objects makes harder sample negative training data via selective search interesting regions object simulate learning situation number labeled real images novel category zero however also experiment small number labeled real images every category randomly select positive training images form datasets sizes final datasets note images contain two positive bounding boxes size virtual dataset noted always images imagenet ilsvrc imagenet network imgnet front front side front side intra front front side front side intra areo aero bike bike bird bird boat boat botl botl bus bus car car cat cat chr chr cow cow tab tab dog dog hse hse mbik pers mbik pers plt plt shp shp sofa sofa trn trn map map table results training different synthetic views cnn used top table trained classification cnn bottom table also finetuned pascal detection net views aero bike bird bus car cow dog hrs mbik shp trn map table results training different real image views represent removing certain view note map subset pascal dataset figure detections amazon domain office showing examples synthetic model second row green bounding box improves localization compared model trained real webcam images first row red bounding box figure detection results proposed vcnn pascal real annotated images limited available novel category vcnn performs much better rcnn fast adaptation method get vcnn network train svm classifiers baselines use datasets train rcnn model train fast adaptation method described rcnn imagenet ilsvrc however detection data limited results results figure show number real training images limited method vcnn performs better traditional rcnn vcnn also significantly outperfoms method based hog features also confirm proposed rrrr data synthesis methodology better simulating background texture partcular virtual data boosts map table without using real training examples real images per category absolute improvement rcnn also notice results much lower unlike results experiment may explained fact selective search generates many without color using regions probably decreases cnn ability recognize realistic color objects note vcnn trained real images per category total also using approximately real images texture background however still training webcam map table detection results proposed vcnn object categories office dataset test data experiments real images amazon domain compare training real training images webcam domain top row model trained representing virtual images uniform gray texture real texture respectively results clearly demonstrate real training data mismatched target domain synthetic training provide significant performance boost detection much fewer annotated bounding boxes pascal training set much easier collect texture images need bounding box annotation yet obtained map comparable map achieved dpm without context rescoring trained full dataset speaks power transferring deep representations suggests synthetic cad data promising way avoid tedious annotation novel categories emphasize significant boost due adapting features synthetic data via showing adapted features better fixed features generation settings results novel domains test images come different visual domain dataset training images expect performance detector degrade due dataset bias experiment evaluate benefit using synthetic cad data improve performance novel domains use part office dataset categories common objects cups keyboards etc domain amazon images target testing domain downloaded webcam images training domain collected office environment generate synthetic data categories office dataset downloaded roughly five models category data generation method experiments pascal expcept use original texture models experiment considering texture objects office dataset simpler compare two generation settings representing virtual images uniform gray texture real texture respectively background settings white match majority amazon domain backgrounds generate images model producing images total use synthetic images train vcnn deep detector test amazon domain images baseline train baseline deep detector webcam domain total images also test images amazon domain results results shown table mean vcnn versus deep detector trained webcam domain significant boost performance setting considerably worse shows potential synthetic cad training dataset bias scenarios figure show examples object detected detector trained webcam detected perfectly vcnn model obtain results selected bounding box highest score region proposals image conclusion paper demonstrated synthetic cad training modern deep cnns object detectors successful training data novel objects domains limited investigated sensitivity convnets various cues training data pose foreground texture color background image color simulate factors used synthetic data generated cad models results demonstrated popular deep convnet detection task indeed largely invariant cues training synthetic images simulated cues lead similar performance training synthetic images without cues however network task invariance diminished thus novel categories adding synthetic variance along dimensions layers proved useful based findings proposed new method learning object detectors new categories avoids need costly image annotation advantageous one needs learn detector novel object category instance beyond available labeled datasets also showed method outperforms detectors trained real images real training data comes different domain one case domain shift findings preliminary experiments domains necessary acknowledgements thank trevor darrell judy hoffman anonymous reviewers suggestions work supported nsf award references berg deng imagenet large scale visual recognition challenge deng dong socher feifei imagenet hierarchical image database computer vision pattern recognition cvpr ieee conference pages ieee everingham van gool williams winn zisserman pascal visual object classes voc challenge international journal computer vision june felzenszwalb girshick mcallester ramanan object detection discriminatively trained partbased models pattern analysis machine intelligence ieee transactions girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation arxiv preprint jaderberg simonyan vedaldi zisserman synthetic data artificial neural networks natural scene text recognition workshop deep learning nips krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips lecun boser denker henderson howard hubbard jackel backpropagation applied handwritten zip code recognition neural computation lenc vedaldi understanding image representations measuring equivariance equivalence corr liebelt schmid object class detection geometric model computer vision pattern recognition cvpr ieee conference liebelt schmid schertler viewpointindependent object class detection using feature maps computer vision pattern recognition cvpr ieee conference lin maire belongie hays perona ramanan zitnick microsoft coco common objects context eccv mahendran vedaldi understanding deep image representations inverting arxiv malisiewicz gupta efros ensemble object detection beyond computer vision iccv ieee international conference pages ieee nevatia binford description recognition curved objects artificial intelligence pinto barhomi cox dicarlo comparing visual features invariant object recognition tasks applications computer vision wacv ieee workshop pages ieee rematas ritschel fritz tuytelaars imagebased synthesis viewpoints guided models ieee conference computer vision pattern recognition cvpr oral saenko kulis fritz darrell adapting visual category models new domains computer eccv pages springer sermanet eigen zhang mathieu fergus lecun overfeat integrated recognition localization detection using convolutional networks corr simonyan zisserman deep convolutional networks image recognition corr stark goesele schiele back future learning shape models cad data proc bmvc pages sun saenko virtual reality fast adaptation virtual object detectors real domains bmvc sun savarese probabilistic model object classes computer vision pattern recognition cvpr ieee conference van sande gevers smeulders selective search object recognition ijcv xiang mottaghi savarese beyond pascal benchmark object detection wild ieee winter conference applications computer vision wacv yosinski clune bengio lipson transferable features deep neural networks ghahramani welling cortes lawrence weinberger editors advances neural information processing systems pages zeiler fergus visualizing understanding convolutional networks arxiv
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cohomological characterization locally virtually cyclic groups sep dieter degrijse abstract show countable group locally virtually cyclic bredon cohomological dimension family virtually cyclic subgroups one introduction let discrete group recall geometric dimension smallest possible dimension free contractible cohomological dimension definition shortest length projective resolution group ring cohomological dimension algebraic counterpart geometric dimension satisfies max classic construction eilenberg ganea one clearly trivial standard exercise check equivalent important result says equivalent free group hence possible difference might exist group open question known problem whether group exists one generalize setup free actions actions stabilizers lying certain family subgroups collection subgroups closed conjugation taking subgroups classifying family model stabilizers fixed point sets contractible geometric dimension family denoted gdf definition smallest possible dimension model geometric dimension also algebraic counterpart bredon cohomological dimension family denoted cdf bredon cohomology family generalization ordinary group cohomology category replaced category contravariant functors orbit category category abelian groups algebraic invariant cdf defined shortest length projective resolution constant functor category easy check one lets family containing trivial subgroup setup coincides free case described first paragraph shown cdf gdf max cdf date february dieter degrijse moreover easy see gdf fact equivalent cdf follows result symonds lemma seems reach moment say anything relationship cdf gdf general family one say one considers either family finite subgroups fin family virtually cyclic subgroup besides trivial family family finite subgroups family virtually cyclic subgroups studied extensively last years partially due appearance isomorphism conjectures conjectures predict isomorphisms certain equivariant cohomology theories classifying spaces families reduced group group rings see fin numbers cdf gdf denoted respectively note cdq since evaluating projective tensoring result yields projective follows work dunwoody finitely generated groups virtually free context answer problem also known indeed brady leary nucinkis construct certain right angled coxeter groups finally turn family virtually cyclic subgroups case cdf gdf denoted respectively answer eilenbergganea problem also known fluch leary show certain right angled coxeter groups satisfy paper fluch leary also ask prove following main theorem group satisfies locally virtually cyclic proof main theorem finalized section result serre cor implies locally virtually cyclic hand countable locally virtually cyclic groups satisfy hence main theorem implies following corollary countable group one recall right ovc called flat categorical tensor product exact functor category left ovc abelian groups definition constant functor flat bredon homological dimension family virtually cyclic subgroups equals zero since countable see prop section locally virtually cyclic groups satisfy prop main theorem implies following corollary countable group locally virtually cyclic constant functor flat ovc light corollary resist mentioning conjecture saying countable group locally virtually cyclic group von neumann algebra flat cohomological characterization locally virtually cyclic groups let free group generators recall group said coherent finitely generated subgroups finitely presented proof main theorem also show certain groups admit three dimensional model corollary let countable group satisfying one following hold coherent contain moreover every group three dimensional model classifying space virtually cyclic stabilizers proof corollary given end section note follows prop corollary bounds given optimal intermediate values occur motivated remark conjecture countable groups preliminaries start recalling basic notions bredon cohomology cohomology theory introduced bredon finite groups mean develop obstruction theory equivariant extension maps later generalized arbitrary groups applications finiteness conditions see section refer reader details let discrete group let family subgroups orbit category category whose objects left cosets whose morphisms maps objects right contravariant functor category denoted defined objects morphisms natural transformations objects category abelian free objects precisely free objects isomorphic direct sums modules form free basis set sequence modules said exact exact evaluated every object also yoneda lemma allows one construct free projective resolutions similar way case ordinary cohomology follows contains enough projectives construct projective resolutions hence one construct extnof usual properties denote contravariant functor maps fixed points maps morphisms obvious restriction maps bredon cohomology coefficients definition hnf extnof trivial bredon cohomology fixed point functor coefficients understood terms ordinary group cohomology lemma cor one dieter degrijse ker moreover ker bredon cohomological dimension family denoted cdf defined cdf sup hnf standard methods homological algebra one shows number coincides length shortest projective free resolution using version shapiro lemma bredon cohomology one show cdf cdf subgroup remainder paper always denote family finite subgroups key ingredient proof main theorem following construction weiermann see relates bredon cohomology family virtually cyclic subgroups bredon cohomology family finite subgroups let denote set infinite virtually cyclic subgroups two infinite virtually cyclic subgroups said equivalent denoted commensurable meaning intersection finite index using fact two infinite virtually cyclic subgroups virtually cyclic group equivalent see lemma easily seen indeed defines equivalence relation one also verify equivalence relation satisfies following two properties let define group group called commensurator sometimes denoted commg confused normalizer note depends equivalence class contains subgroup define following family subgroups words contains finite subgroups infinite virtually cyclic subgroups equivalent pushout diagram theorem yields following see also cohomological characterization locally virtually cyclic groups proposition notation let denote set equivalence classes let complete set representatives orbits conjugation action every exists long exact sequence hivc hif hif hif end section simple observations groups recall group group presentation group solvable definition ascending lemma group following hold group contain copy group group contain copy free group two generators every right ovc isomorphism proof since corollary lemma follows contain copies groups proves part immediate proposition lemma finitely generated group cdq free proof follows cor cdq cdq virtually free dunwoody result mentioned introduction however since contain lemma conclude virtually cyclic implying impossible conclude cdq since contain virtually cyclic follows stalling theorem ends groups finitely generated implies free relative ends let finitely generated group let infinite index finitely generated subgroup commutative ring unit one defines points cokernel natural inclusion right indg coinde coindh coinde coinde dieter degrijse considering long exact cohomology sequence associated short exact sequence right coindg coinde application shapiro lemma yields exact sequence kropholler roller define number ends pairs number field two elements note equals trivial subgroup coincides number ends mentioned previous section subset said resp resp complement covered finitely many right cosets subset said proper neither hfinite subset called invariant symmetric difference shall see later importance proper invariant subsets contained fact certain additional hypotheses used construct trees note coindg corresponds set subsets equipped symmetric difference one easily checks subsets exactly subsets correspond elements coindg elements finally one verifies proper invariant subsets map element map correspond elements appearing observations lead following lemma lemma lemma proper invariant subset proof consider commutative diagram exact rows coindg coindg vertical maps restriction maps induced inclusion double restricted isomorphic coset formula says ghg indh coinde since finite index follows isomorphic direct sum modules coinduced trivial group since finitely generated implies proper exist element almost invariant subset diagram conclude must contained image means cohomological characterization locally virtually cyclic groups next section apply lemma however due presence torsion groups forced consider cohomology coefficient rationals therefore need argue one also compute instead show follow approach swan lemma proven fix commutative ring unit finite generating set lemma let set maps turned right via let subspace consisting maps take values finitely many right cosets let subspace consisting maps exists finite union hgj proof take let finite union right cosets letting one sees indeed hence proves coindg explicitly natural inclusion right given homr homr otherwise composing inclusion isomorphism right homr gives inclusion homr otherwise shows hence using fact set generators one verifies coincides subspace consisting maps exists finite union lemma follows easily recall rank smallest positive integer admits surjection onto integer exists rank infinite proposition commutative ring unit one proof let cayley graph generating set acts left vertices elements connected edge iff forssome since generates follows connected let hgi union finitely many right cosets mean subgraph spanned vertices since dieter degrijse acts interested connected components fix vertex given exists path cross connected component cross exists vertex generator vertex subpath going cross hence contained connected component follows every connected component contains either element form hgj conclude finitely many connected components enumerate elements let defined connected define union let components covered finitely many right cosets note every finite union right cosets let set components since component contained unique component obtain map construction component covered finitely many right cosets follows map surjective indeed let component know covered finitely many right cosets intersect connected component covered finitely many right cosets therefore suppose intersects consider connected components suppose connected components could covered finitely many right cosets since finitely many connected components follows covered finitely many right cosets contradiction hence must exist connected component finitely covered construction subset element maps set maps surjection let defines monomorphism onto direct summand define map lies component see set pick since connected edge way component possible conclude one checks maps assemble form map lim claim isomorphism indeed take let finite union right cosets large enough implying constant connected components proving surjective suppose hence finite union right cosets proving since follows injective proves claim cohomological characterization locally virtually cyclic groups consider two cases first suppose exists infinite case rkr rkr infinite since direct summand lim commutative ring lemma conclude rkr unit holds particular secondly assume finite case one since conclude taking limits lemma finitely generated free abelian group follows free abelian group countable rank conclude rkr rkz commutative ring unit holds particular rkr desired corollary one dimq proof follows immediately proposition structure commensurators crucial ingredient proving main theorem following variation result due kropholler theorem finitely generated group rational cohomological dimension two infinite virtually cyclic subgroup splits virtually cyclic subgroup commensurable particular contains group theorem allows determine structure commensurators infinite virtually cyclic groups group corollary group commensurator infinite virtually cyclic subgroup locally virtually cyclic proof let infinite virtually cyclic subgroup let finitely generated subgroup contains finitely generated group aim show virtually cyclic assume virtually cyclic case cdq lemma follows theorem contains implies lemma contradiction conclude virtually cyclic dieter degrijse turn proof theorem kropholler proves finitely generated group cohomological dimension two infinite cyclic subgroup splits infinite cyclic subgroup commensurable show kropholler arguments remain valid groups rational cohomological dimension equal two let finitely generated group rational cohomological dimension equal two infinite cyclic subgroup lemma lemma group duality group proof strebel criterion mentioned beginning section lemma valid field even though lemma formulated one easily verifies proof carries field particular follows group cdq type satisfies implies duality group dualising module recall called implies note integral domain lemma lemma one proof proof lemma shown set elements form right ore set meaning every exists using one easily checks set elements also form left ore set meaning every exists let field fractions consider right using left ore condition one checks divisible right since also right follows lemma since type implies let element let left generated let generator since exists element implies generating set space particular finite dimensional space hence prove corollary know dimq lemma need show since duality group since indg coinde follows shapiro lemma duality coindh coinde cohomological characterization locally virtually cyclic groups follows order prove lemma suffices shows coindh module constructed since finite dimensional space isomorphism left homq homq action element homq given also isomorphism left homq homq every left hand side one appearing right hand side right hand side trivial right hand side coindh rese follows coindh coindh resh prove theorem proof since lemma follows lemma proper invariant subset proof lemma carries verbatim showing splits virtually cyclic subgroup commensurable means either equal amalgamated free product subgroups first suppose note equal since amalgamated free product index two fits short exact sequence since isomorphic infinite dihedral group infinite virtually cyclic standard exercise group theory check contains index least three either follows normal form theorem free amalgamated products contains secondly suppose two cases two consider hnn extension ascending determined injective homomorphism since case infinite virtually cyclic hard see one find infinite cyclic subgroup hxi restricts hxi britton lemma shows subgroup ascending hnn extension generated stable letter isomorphic group hand follows britton lemma contains finishes proof turn proof corollary introduction proof let group satisfying follows proposition order prove suffices prove cdf infinite cyclic subgroup lemma suffices show cdf finitely generated subgroup containing first suppose contain case follows proof theorem either amalgam hnn extension edge vertex groups dieter degrijse infinite virtually cyclic subgroups commensurable case point stabilizers action associated tree commensurable implying model conclude cdf desired next assume coherent implying finitely presented actually need case arguments proof theorem carry completely using finite presentability one shows iterating theorem must eventually produce splitting virtually cyclic subgroups proving acts simplicially tree infinite virtually cyclic vertex groups edge stabilizer commensurable conclude model hence cdf desired finally suppose case follows lemma type argument previous case applies suppose group torsion see prop implies remark follows proof order show every countable group satisfies suffices show finitely generated group infinite virtually cyclic subgroup type able prove type proof main theorem let finitely generated group aim show virtually cyclic argue contradiction end assume follows lemma cdq free moreover corollary commensurator infinite virtually cyclic subgroup locally virtually cyclic implies family coincides family hence cdf follows lemma right ovc holds particular every fixed point functor let subset containing virtually cyclic since finite groups rational cohomological dimension zero conclude lemma every since cdq deduce fix denote recall free see lemma consider short exact sequence cohomological characterization locally virtually cyclic groups augmentation map since long exact cohomology sequence associated short exact sequence implies inducting obtain short exact sequence since long exact cohomology sequence associated short since exists short exact sequence implies exact sequence free inducting obtain short exact sequence free since every long exact cohomology sequence associated yields exact sequences every obtain commutative diagram exact rows conclude since direct summand summand conclude direct implies finite index since finitely generated contradiction conclude virtually cyclic main theorem follows easily group every finitely generated subgroup also satisfies therefore virtually cyclic proves locally virtually cyclic acknowledgment author partially supported danish national research foundation centre symmetry deformation author also grateful centro ciencias unam morelia particular hospitality visit part work completed dieter degrijse references brady leary nucinkis algebraic geometric dimensions groups torsion lond math soc vol bredon equivalent cohomology theories lecture notes mathematics springer degrijse petrosyan commensurators classifying spaces virtually cyclic stabilizers groups geometry dynamics vol degrijse petrosyan geometric dimension groups family virtually cyclic subgroups journal topology degrijse petrosyan bredon cohomological dimensions groups acting cat groups geometry dynamics degrijse petrosyan cohomology appl categor struct dembegioti petrosyan talelli intermediaries bredon homology classifying spaces publicacions dunwoody accessibility groups cohomological dimension one london math soc eilenberg ganea category abstract groups ann math fluch bredon homological dimensions groups thesis university southampton fluch leary phenomenon actions virtually cyclic stabilizers groups geometry dynamics fluch nucinkis classifying space family virtually cyclic subgroups elementary amenable groups proc amer math soc kropholler groups groups cohomological dimension two commentarii mathematici helvetici vol kropholler roller relative ends duality groups journal pure applied algebra vol leary classifying spaces family virtually cyclic subgroups recent developments alg vol contemp math soc survey classifying spaces families subgroups infinite groups geometric combinatorial dynamical aspects springer theory applications geometry vol series modern surveys mathematics berlin meintrup universal space group actions compact isotropy proc conference geometry topology aarhus reich conjectures handbook vol springer berlin weiermann classifying space family virtually cyclic subgroups pure applied mathematics quarterly vol lyndon schupp combinatorial group theory springer spectral sequence bredon homology pure appl algebra nucinkis dimensions bredon homology homology homotopy applications vol serre trees springer stallings groups dimension locally free bull amer math soc swan groups cohomological dimension one algebra symonds bredon cohomology subgroup complexes pure appl algebra school mathematics statistics applied mathematics nui galway ireland address
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dec kalman filter modern extensions nonlinear filtering problem amirhossein taghvaei department mechanical science engineering university illinois email jana wiljes institut mathematik potsdam email wiljes prashant mehta department mechanical science engineering university illinois email mehtapg sebastian reich institut mathematik potsdam department mathematics statistics university reading email sereich paper concerned filtering problem three algorithmic solution approaches problem reviewed classical filter provides exact solution linear gaussian problem ensemble filter enkbf approximate filter represents extension filter nonlinear problems iii feedback particle filter fpf represents extension enkbf furthermore provides consistent solution general nonlinear case common feature three algorithms gain times error formula implement update step account conditioning due observations filter contrast commonly used sequential monte carlo methods enkbf fpf avoid resampling particles importance sampling update step moreover feedback control structure provides error correction potentially leading smaller simulation variance improved stability properties paper also discusses issue filter update formula formulates novel approximation algorithm based ideas optimal transport coupling measures performance algorithms illustrated numerical example introduction since pioneering work kalman sequential state estimation extended application areas far beyond original aims numerical weather prediction oil reservoir exploration history matching developments made possible clever combination monte carlo techniques kalmanlike techniques assimilating observations underlying dynamical models prominent algorithms ensemble kalman filter enkf randomized maximum likelihood rml method unscented kalman filter ukf invented independently several research groups enkf particular viewed cleverly designed random dynamical system interacting particles able approximate exact solution relative small number particles interacting particle perspective led many new filter algorithms recent years beyond inherent gaussian approximation enkf data assimilation update step paper review interacting particle perspective context filtering problems demonstrate close relation kalman bucy kbf filter equations ekbf extended filter equations stochastic ensemble filter equation deterministic ensemble filter equation feedback particle filter equation enkbf fpf table nomenclature filtering algorithms original feedback control structure data assimilation step specifically highlight feedback control structure three classes algorithms approximating posterior distribution classical filter provides exact solution linear gaussian problem ensemble filter enkbf approximate filter represents extension filter nonlinear problems iii feedback particle filter fpf represents extension enkbf furthermore provides consistent solution general nonlinear problem closely related goal provide comparison algorithms common feature three algorithms gain times error formula implement update step filter difference filter exact algorithm two algorithms approximate error decreasing zero number particles increases infinity algorithms property said consistent class interacting particle algorithms discussed fpf represents general solution nonlinear filtering problem challenge implementing fpf lie approximation gain function used update step gain function equals kalman gain linear gaussian setting must numerically approximated general setting one particular approximation constant gain approximation case fpf shown reduce enkbf algorithm enkbf naturally extends nonlinear dynamical systems versions become popular recent years applications example dynamics oil reservoir exploration setting development application closely related particle flow algorithms also subject recent interest outline remainder paper follows filtering problem classic filter summarized sections respectively filter put context interacting particle systems form enkbf section section together appendices provides consistent definition fpf discussion alternative approximation techniques lead consistent approximations filtering problem number particles goes infinity shown enkbf viewed approximation fpf four mic approaches gain function approximation described relationship discussed performance four algorithms numerically studied compared example problem section paper concludes discussion open problems section nomenclature filtering algorithms described paper appears table problem statement setting model nonlinear filtering problem described nonlinear stochastic differential equations sdes signal observation dxt dbt dzt dwt hidden state time initial condition sampled given prior density observation measurement vector two mutually independent wiener processes taking values mappings known functions covariance matrix observation noise assumed positive definite function column vector whose coordinate denoted scaling assumed without loss generality covariance matrices associated identity matrices unless otherwise noted stochastic differential equations sde expressed form table includes list symbols used filtering problem applications continuous time filtering models often expressed dxt dzt mutually independent white noise processes gaussian noise vector valued observation time model preferred mathematical rigor sde involving variable notation model variable notn defn state cond mean process noise wiener process cond var measurement kalman gain measurement noise wiener process table symbols filtering problem converted ode involving formally dividing sde replacing see also remark objective filtering estimate posterior distribution given time history observations density posterior distribution denoted measurable set one example particular interest mappings linear constant matrix depend upon prior density gaussian associated problem referred linear gaussian filtering problem problem posterior density known gaussian resulting filter said finitedimensional posterior completely described finitely many statistics conditional mean variance linear gaussian case general nonlinear case however filter defines evolution space probability measures particle filter algorithm approximate posterior key step construction interacting stochastic processes xti value xti state particle time time empirical distribution formed particle population used approximate posterior distribution recall defined measurable set xti indicator function equal otherwise first interacting particle representation filtering problem found close connection interacting particle formulations gain factor innovation structure classic kalman filter made explicit starting led fpf formulation considered paper notation density gaussian random variable mean variance denoted vectors model htt table symbols kalman filter dot product denoted denotes transpose vector similarly matrix denotes matrix transpose two sequence big notation means filter consider linear gaussian problem mappings matrices process noise covariance constant matrix prior density gaussian denoted problem posterior density known gaussian denoted conditional mean variance evolution described filter dzt htt referred kalman gain filter initialized initial conditions prior density table includes list symbols used kalman filter evolution equation mean sde presence stochastic forcing term side evolution equation variance ode depend upon observation process kalman filter one widely used algorithm engineering although filter describes posterior linear gaussian settings often used approximate algorithm even general settings defining matrices according jacobians mappings resulting algorithm referred extended kalman filter dzt used formula gain kalman filter extensions recursive algorithms process measurements sequential online fashion time filter computes error dzt called innovation error reflects new information contained recent measurement filter state corrected time step via gain error update formula error correction feedback structure see fig important account robustness filter based idealized model underlying stochastic dynamic process property feedback provides robustness allowing one tolerate degree uncertainty inherent model simple intuitive nature update formula invaluable design testing operation filter example kalman gain proportional scales ratio measurement model practice gain may tuned optimize filter performance minimize online computations offline solution algebraic ricatti equation obtained equating side variance ode zero may used obtain constant value gain basic kalman filter also extended handle filtering problems involving additional uncertainties signal model observation model resulting approximate algorithms referred interacting multiple model imm filter probabilistic data association pda filter respectively pda filter gain varies based estimate instantaneous uncertainty measurements imm filter multiple kalman filters run parallel outputs combined obtain estimate one explanation feedback control structure kalman filter based duality estimation control although limited linear gaussian problems considerations also help explain differential ricatti equation structure variance ode although widely used extended kalman filter suffer stability issues crude approximation nonlinear model observed divergence arises account two reasons even gaussian process measurement noise nonlinearity mappings lead forms posterior density jacobians used propagating covariance lead large errors approximation gain particularly hessian mappings large issues necessitated development particle based algorithms described following sections variable notation particle state xti empirical variance model stoch enkbf deter enkbf particle process noise bti wiener process particle meas noise wti wiener process table symbols ensemble filter ensemble filter pedagogical reasons ensemble filter enkbf best described linear gaussian problem also approach taken section extension nonlinear problem immediate similar extension kalman filter extended kalman filter even linear gaussian settings particle filter may computationally efficient option problems large state dimension weather models meteorology large computational bottleneck simulating kalman filter arises due propagation covariance matrix according differential riccati equation computation scales memory enkbf implementation one replaces exact propagation covariance matrix empirical approximation particles xti computation scales reduction computational cost achieved reduced rank kalman filter however connection empirical measures crucial application enkbf nonlinear dynamical systems enkf algorithm first developed discretetime setting since various formulations enkf proposed state two continuoustime formulations enkbf table includes list symbols used formulations stochastic enkbf conceptual idea stochastic enkbf algorithm introduce zero mean perturbation noise term innovation error achieve consistency variance update stochastic enkbf algorithm particles evolve according dxti axti dbti dzti hxti dwti xti state ith particle time initial condition bti standard wiener process wti standard wiener process assumed independent bti variance obtained empirically using note particles interact common covariance matrix idea introducing noise process first appeared enkf derivation continuoustime stochastic enkbf found based limiting argument whereby update step formally viewed discretization stochastic sde linear gaussian problem stochastic enkbf algorithm consistent limit means conditional distribution xti gaussian whose mean variance evolve according kalman filter equations respectively update formula used unique deterministic analogue described next deterministic enkbf deterministic variant enkbf first proposed given dxti axti dbti hxti dzt proof consistency deterministic variant enkbf linear systems found close parallels deterministic enkbf fpf explored section deterministic formulation enkbf interaction particles arises covariance matrix mean although stochastic deterministic enkbf algorithms consistent linear gaussian problem easily extended nonlinear settings however resulting algorithm general consistent accuracy enkbf recent research enkf focussed long term behavior accuracy filters applicable nonlinear data assimilation particular mathematical justification feasibility enkf continuous counterpart small ensemble limit interest studies accuracy finite ensemble exceptional importance due fact large number ensemble members option computational point view many applicational areas authors show asymptotic accuracy limit particular type variance inflation deployed stochastic enkbf discrete continuous formulation enkf also shown similar results concerning accuracy deterministic variant enkbf derived fully observed system assumed deriving accuracy results investigation partially observed systems particularly relevant update step cases cause divergence estimate sense signal lost values estimate reach machine infinity feedback particle filter fpf controlled sde dxti xti xti dbti xti dzt xti update similar enkbf xti state ith particle time initial condition bti standard wiener process xti bti mutually independent also independent update term indicates sde expressed stratonovich form gain function dimension needs obtained fixed time gain function defined solution pde introduced following subsection linear gaussian problem kalman gain general nonlinear gain function needs numerically approximated algorithms also summarized following subsection remark given stratonovich form provides mathematical interpretation formal ode model see section sde textbook also tain formal ode model filter denoting ode model filter white noise process given dxti feedback particle filter thus provides generalization kalman filter nonlinear systems innovation feedback structure control preserved see fig linear gaussian case gain function kalman gain nonlinear case kalman gain replaced nonlinear function state see fig remark shown appendix condition gain function computed exactly fpf exact algorithm initial condition sampled prior xti fig innovation feedback structure kalman filter nonlinear feedback particle filter variable notation model particle state xti fpf gain function poisson gain function time given constant gain particle gain xti galerkin optimal coupling remark gain function uniquely defined filtering problem formula represents one choice gain function generally sufficient require solution table symbols feedback particle filter numerical implementation finite number particles simulated xti xti law large numbers lln considerations appendix described general setting applicable stochastic processes xti evolving manifolds also explains update formula stratonovich form sdes manifold well known stratonovich form invariant coordinate transformations intrinsic ito form discussion fpf lie groups appears table includes list symbols used fpf gain function pedagogical reasons primarily notational convenience gain function defined case case gain function defined terms solution weighted poisson equation denote gradient divergence operators respectively time denotes density xti terms solution extension observation straightforward appears although paper limited proposed algorithm applicable nonlinear filtering problems differential manifolds matrix lie groups intrinsic form poisson equation see domains boundary pde accompanied neumann boundary time one justification choosing gradient form solution optimality general solution given solution solves easy see therefore minimum solution interpreting norm kinetic energy gain function defined seen optimal sense optimal transportation alternative solution provided definition leads standard poisson equation unknown potential fundamental solution explicitly known fact exploited interacting particle filter representations two special cases summarized part following two examples exact solution found condition boundary domain unit normal vector boundary point fig exact solution poisson equation using formula density sum two gaussians density depicted shaded curve background example scalar case poisson equation constant gain approximation constant gain approximation best sense constant approximation gain function see figure mathematically obtained considering following optimization problem arg min using standard sum squares argument multiplying sides pde integrating parts expected value computed explicitly integrating yields solution explicitly fig constant gain approximation feedback particle filter particular choice sum two gaussians solution obtained using depicted fig problem statement given samples drawn approximate gain function density explicitly known four numerical algorithms approximation gain function appear following four algorithms based theory solution poisson equation pde described integral evaluated empirically obtain following approximate formula gain example suppose density gaussian observation function gain function kalman gain general case solution known explicit form must numerically approximated note even two exact cases one would need numerically approximate solution density available explicit form problem statement numerical approximation follows formula referred constant gain approximation gain function popular choice applications equivalent approximation used deterministic stochastic enkbf example consider linear case constant gain approximation formula equals empirical mean empirical variance respectively linear gaussian case fpf algorithm constant gain approximation gives deterministic enkf algorithm galerkin approximation galerkin approximation generalization constant gain approximation gain function approximated subspace span mathematically galerkin solution defined optimal approximation algorithm constant gain approximation input output calculate calculate arg min algorithm galerkin approximation gain function solution easily obtained applying projection theorem gives input output denoting expressing finite dimensional system expressed linear matrix equation matrix vector whose entries given respective formulae example two types approximations follow consideration two types basis functions constant gain approximation obtained taking basis functions choice identity matrix galerkin gain function constant vector empirical approximation single basis function galerkin solution subspace sobolev space defined space functions respect density whose weak derivatives also respect density appropriate space solution poisson equation calculate solving empirical approximation obtained xti xti xti practice matrix vector approximated empirically equation solved obtain empirical approximation denoted see table galerkin algorithm terms empirical approximation gain function approximated choice basis function problem dependent euclidean settings linear basis functions standard lead constant gain approximation discussed example straightforward extension choose quadratic higher order polynomials basis functions however approach scale well dimension problem number basis functions scales least linearly dimension galerkin algorithm involves inverting matrix motivates nonparametric data driven approaches small number basis functions selected adaptive fashion one algorithm proposed based expansion next two subsections alternate algorithms described one advantage algorithms require selection basis functions approximation linear operator pde generator markov semigroup denoted follows solution equivalently expressed fixed representation useful approximated operator mula succinctly expressed easy verify therefore equals constant gain approximation formula kernel expressed terms gaussian kernel exp normaliza tion factor chosen shown approximation problem obtained algorithm approximation gain function input output calculate exp calculate calculate calculate initialize calculate small method successive approximation used solve equation recursive simulation algorithm initialized solution previous gain function obtained taking gradient two sides purpose useful first define operator terms operator gain function approximated gil ikj calculate end calculate calculate til calculate optimal approximation optimal approximation another nonparametric approach directly approximate gain function ensemble algorithm presented first time context fpf approximation based upon reformulation recently developed ensemble transform optimally transporting coupling measures relationship gain function approximation follows define family densities sufficiently small consider optimal transport problem objective solution denote constraints min solution problem denoted referred optimal transport map shown appendix algorithm optimal coupling approximation gain function input output calculate calculate calculate solving linear program calculate calculate ensemble transform algorithm approximate solution optimal transportation problem given samples drawn problem gain function approximation algorithm involves first solving linear program objective min constraints solution denoted referred optimal coupling coupling constants interpretation joint probabilities two equality constraints arise due specification two marginals noted particles sampled optimal value approximation optimal value objective latter celebrated wasserstein distance terms optimal solution linear program approximation gain function obtained galerkin algorithm polynomial basis functions case gives constant gain approximation algorithm kernel algorithm optimal coupling algorithm first numerical experiment fixed number particles drawn figures depicts approximate gain function obtained using three algorithms ease comparison exact solution also depicted second numerical experiment empirical error evaluated function number particles single simulation error defined according error numerics section contains results numerical experiments algorithms applied bimodal distribution problem introduced example density mixture two gaussians exact solution obtained using explicit formula depicted fig following parameters used numerical implementation algorithms dirac delta tensor otherwise practice finite appropriately chosen approximation becomes exact approximation structurally similar constant gain approximation formula also approximation formula three cases gain ith particle approximated linear combination particle states approximations computationally attractive whenever dimension state space high dynamics confined subset however priori known kex particles kalg output algorithm kex exact gain estimate error evaluated averaging simulations simulation new set particles sampled used input consistently three algorithms figure depict estimate error function number particles third numerical experiment effect varying parameter investigated optimal coupling algorithms experiment fixed number particles used figure depict estimate error function parameter following observations made based results numerical experiments figure accuracy galerkin algorithm improves number basis function increases fixed number particles matrix becomes poorly conditioned number basis functions becomes large lead numerical instabilities solving matrix equation figure optimal coupling algorithms preserve positivity property exact gain positivity property gain necesarily preserved galerkin algorithm correct sign gain important filtering applications gain determines direction drift particles wrong sign lead divergence particle trajectories exact error galerkin gain approxmiation galerkin gain approx error function error gain approximation gain approx error function error optimal coupling gain approximation optimal coupling gain approx error function fig comparison gain function approximations obtained using galerkin part kernel part optimal coupling part algorithms exact gain function depicted solid line density depicted shaded region background parts depict estimate empirical error function number particles estimate obtained averaging empirical error simulations figure fixed number particles optimal value minimizes error kernelbased optimal coupling algorithms algorithm shown small large error scales approximate gain converges constant gain approximation particular error remains bounded even large values optimal coupling algorithm leads spatially irregular approximations nevertheless converge number particles increases optimal choice parameter depends particle size finally spatially regular approximation could obtained using kernel dressing convoluting particle approximation gaussian kernels conclusion summarized interacting particle representation classic filter extension variance bias dominates dominates fig comparison estimate empirical error function parameter number particles nonlinear systems distributions general framework fpfs framework attractive since maintains key structural elements classic filter namely gain factor innovation particular enkf become widely used data assimilation dynamics oil reservoir exploration robust extensions enkf distributions urgently needed fpfs provide systematic approach extensions spirit kalman original work however interacting particle representations come price require approximate solutions elliptic pde coupling problem viewed probabilistic perspective hence robust efficient numerical techniques interacting particle systems study behavior primary focus future research acknowledgement first author supported part computational science engineering cse fellowship university illinois uiuc research pgm uiuc additionally supported national science foundation nsf grants support gratefully acknowledged research jdw partially funded deutsche forschungsgemeinschaft dfg grant crc scaling cascades complex systems project multiscale data asymptotic model assimilation atmospheric flows grant crc data assimilation project stability accuracy ensemble transform filter algorithms references kalnay atmospheric modeling data assimilation predictability cambridge university press oliver reynolds liu inverse theory petroleum reservoir characterization history matching cambridge university press cambridge kitanidis geostatistical theory inversion water resources research burgers van leeuwen evensen analysis scheme ensemble kalman filter mon wea houtekamer mitchell sequential ensemble kalman filter atmospheric data assimilation mon wea julier uhlmann new extension kalman filter nonlinear systems signal processing sensor fusion target recognition conference vol reich cotter probabilistic forecasting bayesian data assimilation cambridge university press cambridge daum huang noushin exact particle flow nonlinear filters spie defense security sensing daum huang noushin generalized gromov method stochastic particle flow filters spie security international society optics photonics coleman generalizing posterior matching scheme higher dimensions via optimal transportation communication control computing allerton annual allerton conference moselhy marzouk bayesian inference optimal maps journal computational physics heng doucet pokern gibbs flow approximate transport applications bayesian computation arxiv yang blom mehta time feedback particle filter american control conference acc ieee crisan xiong numerical solutions class spdes bounded domains esaim crisan xiong approximate representations class spdes stochastics yang mehta meyn feedback particle filter coupling decision control european control conference ieee conference yang mehta meyn feedback particle filter ieee trans automatic control october blom continuous time roots interacting multiple model filter proc ieee conf decision control daum huang probabilistic data association filter ieee control syst dec kalman new approach linear filtering prediction problems journal basic engineering law stuart zygalakis data assimilation mathematical introduction springerverlag new york reich dynamical systems framework intermittent data assimilation bit numerical analysis bergemann reich ensemble filter continuous data assimilation meteorolog zeitschrift wiljes reich stannat stability accuracy ensemble filter fully observed processes small measurement noise tech https university potsdam del moral kurtzmann tugaut stability uniform propagation chaos extended ensemble filters tech inria bordeaux research center kelly law stuart wellposedness accuracy ensemble kalman filter discrete continuous time nonlinearity kelly stuart ergodicity accuracy optimal particle filters bayesian data assimilation tech https courant institute mathematical sciences kelly majda tong concrete ensemble kalman filters rigorous catastrophic filter divergence pnas oksendal stochastic differential equations introduction applications springer science business media zhang taghvaei mehta feedback particle filter matrix lie groups american control conference acc ieee yang laugesen mehta meyn multivariable feedback particle filter automatica villani topics optimal transportation american mathematical soc evans partial differential equations mass transfer current developments mathematics laugesen mehta meyn raginsky poisson equation nonlinear filtering siam control stano tilton estimation parameters hopper sedimentation model fpf versus bpf control engineering practice tilton ghiotto mehta comparative study nonlinear filtering techniques information fusion fusion international conference berntorp feedback particle filter application evaluation int conf information fusion washington berntorp grover gain computation feedback particle filter american control conference acc ieee coifman lafon diffusion maps appl comput harmon hein audibert luxburg graph laplacians convergence random neighborhood graphs mach learn jun taghvaei mehta meyn error estimates kernel gain function approximation feedback particle filter american control conference acc ieee xiong introduction stochastic filtering theory vol oxford graduate texts mathematics oxford university press exactness feedback particle filter objective section describe consistency result feedback particle algorithm sense posterior distribution particle exactly matches posterior distribution limit put mathematical framework let denote conditional distribution given history filtration observations realvalued function define action according time evolution described kushnerstratonovich pde see theorem dzs smooth functions compact support next define conditional distribution xti given action function defined state space paper considerations section also apply differential manifold proof using sufficient show following two identities according xti time evolution described fokkerplanck equation see proposition dzs gain function control function follows gain function let solution weak form poisson equation control function function gain function requires additional assumptions distribution function assumption probability distribution admits spectral gap functions last step weak form poisson equation used concludes second identity hence theorem relationship gain function optimal transport map let optimal transport map solution problem known map gradient form following furthermore assume map parameter test function assumptions poisson equation unique solution resulting control gain function admissible consistency feedback particle filter stated following theorem theorem let satisfy forward equations respectively assuming functions sobolev space functions squareintegrable respect density whose weak derivatives also respect density assumption function space functions respect first identity obtained using weak form poisson equation second identity obtained similarly use expression control function obtain gain function therefore weak form poisson equation therefore
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ieee transactions ultrasonics ferroelectrics frequency control vol december adaptive vibration suppression system iterative control law piezoelectric actuator shunted negative capacitor pavel jan institute mechatronics computer engineering technical university liberec liberec czech republic research centre special optics optoelectronic systems toptec turnov czech republic feb ceramics laboratory swiss federal institute technology epfl lausanne switzerland dated january abstract adaptive system suppression vibration transmission using single piezoelectric actuator shunted negative capacitance circuit presented known using negative capacitance shunt spring constant piezoelectric actuator controlled extreme values zero infinity since value spring constant controls force transmitted elastic element possible achieve reduction transmissibility vibrations piezoelectric actuator reducing effective spring constant narrow frequency range broad frequency range vibration isolation systems analyzed modeled experimentally investigated problem high sensitivity vibration control system varying operational conditions resolved applying adaptive control circuit parameters negative capacitor control law based estimation value effective spring constant shunted piezoelectric actuator presented adaptive system achieves negative capacitor parameters presented shown arrangement allows design simple electronic system however offers great vibration isolation efficiency variable vibration conditions keywords piezoelectric actuator vibration transmission suppression piezoelectric shunt damping negative capacitor elastic stiffness control adaptive device electronic address introduction vibration suppression nowadays regarded important issue many technical fields ranging robust aerospace industry delicate nanotechnology unsuppressed vibrations traditionally key source noise pollution aging mechanical components even fatal failures transport vehicles machining instruments etc also nanotechnology progress relies high precision devices essentially require efficient vibration control contemporary vibration control techniques based mostly passive methods using elements viscoelastic dampers springs conventional active feedback control principles passive methods rather inexpensive require external source energy bulky inefficient low frequencies hand conventional active methods achieve excellent efficiency subtle device expense high technical complexity high costs lower reliability necessarily gap conventional passive active vibration control methods would balance advantages approaches especially high efficiency also low frequencies low cost promising new approach emerged called methods heralded dawn new era vibration control methods however happen yet suggested methods remain confined research laboratories article present study vibration suppression method uses piezoelectric bulk actuator vibration suppression effect achieved first inserting piezoelectric actuator vibrating structure object isolated vibrations second connecting piezoelectric actuator active external shunt circuit controls effective elastic stiffness actuator aforementioned method suppression vibration transmission introduced hagood von flotow later named piezoelectric shunt damping psd last two decades extensive amount work published psd method examples mentioned passive active broadband adaptive noise vibration control devices vast majority mentioned publications many others classical control theory used description analysis noise vibration suppression systems alternative approach offered date discovered effect shunt circuit mechanical response piezoelectric actuator explained change effective elastic properties piezoelectric actuator actuator inserted source vibrations object isolated vibrations resonant frequency transmissibility vibrations resulting system depends spring constant actuator mass object reduction piezoelectric actuator spring constant results reduction resonant frequency transmissibility vibrations frequencies therefore physics lying behind vibration suppression effect psd method principle passive methods top psd method enables efficient vibration suppression low frequencies principal change concept offers use alternative simpler design tools development noise vibration suppression devices design vibration control systems reduced first study elasticity effect noise vibration transmission second realization active elasticity control early applications followed simple approach demonstrated great potential method stems simplicity noise control system consists selfsensing piezoelectric actuator connected active shunt circuit implementation active shunt circuit electronics using simple analog circuit single linear power amplifier allows significant reduction electric power consumption iii broad frequency range khz system efficiently suppress vibrations despite potential advantages psd method high sensitivity low stability real systems currently prevents industrial exploitation often requires system robustness varying operational conditions actually shown sluka optimal working point pds system lies edge system stability later stability several psd method implementations compared work preumont partial elimination drawbacks achieved adaptive psd vibration control systems reported refs however strong limitation stable vibration isolation efficiency could achieved narrow frequency range aforementioned issues motivated work presented address design adaptive vibration control device principle vibration suppression device presented sec sec iii demonstrate advantages drawbacks narrow broad frequency range vibration isolation device manually adjusted negative capacitor design adaptive vibration isolation device presented sec addition present previously unpublished details control law used obtain results presented refs conclusions experiments presented sec principle vibration suppression known vibration transmission interface two solid objects mainly controlled ratio mechanical impedances since mechanical impedance proportional material stiffness extremely soft element placed two objects works interface high transmission loss vibrations following subsection present simple theoretical model explains effect elasticity mechanical system transmission vibrations system later present method control elastic properties piezoelectric actuator using shunt electric circuit profitably used vibration isolation system effect spring constant transmissibility vibrations scheme vibration isolation measurement system shown fig vibration damping element spring constant damping coefficient placed shaker object mass going isolated vibrations incident transmitted vibrations displacement amplitudes respectively measured using accelerometers transmissibility vibrations considered vibration isolation system defined ratio transmitted incident displacement amplitudes reference source point transmissibility vibration function material parameters control dynamic response mechanical system dynamic response system governed accelerometer piezoelectric actuator mass damping element keff accelerometer negative capacitor znc shaker fig scheme vibration isolation measurement system vibration damping element spring constant damping coefficient placed shaker mass going isolated vibrations incident vibrations displacement amplitude transmitted vibrations displacement amplitude measured using accelerometers vibration damping element used work piezoelectric actuator impedance shunted negative capacitor impedance znc following equation motion considering simplest case transmission harmonic vibrations angular frequency solution yields formula symbols stand mechanical quality factor resonance frequency seen smaller value spring constant smaller value resonant frequency smaller value transmissibility harmonic vibrations angular frequency method active control piezoelectric actuator elasticity figure shows vibration damping element used work piezoelectric actuator capacitance shunted negative capacitor capacitance system example called active elasticity control method introduced date effective spring constant piezoelectric actuator keff derived constitutive equations charge change length piezoelectric actuator appended formula voltage applied back piezoelectric actuator shunt circuit capacitance symbols stand piezoelectric coefficient capacitance spring constant mechanically free piezoelectric actuator respectively combining eqs use relationship capacitance impedance capacitor one readily obtain formula effective spring constant piezoelectric actuator connected external shunt circuit electric impedance keff electromechanical coupling factor piezoelectric element electric impedance mechanically free piezoelectric actuator follows complex impedance shunt circuit approaches value effective spring constant keff piezoelectric element reaches zero figure shows electrical scheme piezoelectric actuator shunted active circuit effectively works negative capacitance referenced negative capacitor effective impedance negative capacitor shown fig equal output voltage gain operational amplifier negative capacitor piezoelectric actuator adjustable resistors fig electrical scheme piezoelectric actuator shunted negative capacitor negative capacitor designed using simple circuit operational amplifier feedback loop use adjustable resistors possible adjust real imaginary part capacitance matches capacitance piezoelectric actuator except sign impedance called reference capacitance negative capacitor approximate formula side stands ideal operational amplifier goes infinity known real imaginary parts piezoelectric actuator capacitance practically depend frequency frequency range resonance frequency situation capacitance piezoelectric actuator approximated high accuracy formula tan tan real part loss tangent piezoelectric actuator capacitance impedance piezoelectric actuator equal tan tan convenient approximate frequency dependence piezoelectric actuator impedance frequency dependence connection capacitor resistor capacitance resistance respectively given critical frequency possible adjust negative capacitor way arg arg situation characterized relation according yields keff effectively reaching zero value transmission vibrations reaches minimum role impedance matching subsection analyze extent condition given eqs must satisfied order achieve required suppression vibration transmission analysis split two steps first analyze sensitivity transmissibility value spring constant actuator second analyze sensitivity effective spring constant actuator keff capacitance negative capacitor order perform first step analysis convenient express suppression level transmissibility vibrations produced active elasticity control using negative capacitor log log transmissibility vibrations given situations shunt circuit connected disconnected piezoelectric actuator respectively small values spring constant frequencies resonant frequency system possible express suppression level transmissibility vibrations form log keff effective spring constant actuator controlled negative capacitor second step convenient denote deviation impedance negative capacitor required value small deviations possible approximate formula keff impedance phase rad transmissibility negative capacitor model experiment model experiment model adjustment model adjustment piezoelectric actuator negative capacitor adjustment negative capacitor adjustment negative capacitor adjustment piezoelectric actuator negative capacitor adjustments khz khz khz impedance abs value imaginary part real part adjustment adjustment adjustment frequency frequency fig frequency dependences physical quantities controls value transmissibility vibrations piezoelectric actuator shunted negative capacitor shown fig shows comparison measured values transmissibility vibrations electrically free piezoelectric actuator filled circles piezoelectric actuator shunted negative capacitor adjusted frequency khz empty circles measured values transmissibility vibrations compared calculated ones theoretical model absolute value electric impedance piezoelectric actuator measured negative capacitor three different adjustments resistors calculated phase electric impedance piezoelectric actuator measured negative capacitor calculated subtracted calculated real imaginary parts effective spring constant piezoelectric actuator shunted negative capacitor estimated decrease level transmissibility vibrations requires decrease effective value spring constant factor considering values electromechanical coupling factor conventional piezoelectric ceramics one conclude relative deviation negative capacitor impedance required value must smaller narrow region capacitances negative capacitor required values spring constant achieved imposes high requirements negative capacitor adjustment clear required adjustment negative capacitor achieved fixed values resistors due limited number commercially available values continuously adjustable trimmers must used next two sections presented experimental results acquired vibration isolation systems manual adaptive adjustment negative capacitor iii manual adjustment negative capacitor next subsection present discuss experimental data measured vibration isolation device negative capacitor shown fig narrow frequency range vibration isolation first frequency dependence transmissibility vibrations electrically free piezoelectric actuator actuator disconnected negative capacitor measured frequency range khz result indicated filled circles fig measured frequency dependences transmissibility vibrations compared predictions theoretical formula given values spring constant mass mechanical quality factor piezoelectric actuator obtained using method least squares next step negative capacitor assembled using operational amplifier according scheme shown fig output voltage gain approximated function condition given eqs achieved setting values resistances according following formulae order find proper adjustment negative capacitor frequency dependence electric impedances piezoelectric actuator reference capacitance fig electrical scheme reference impedance inside negative capacitor shown fig narrow frequency range broad frequency range vibration isolation system measured using spectrum analyzer shown figs using least squares method following values obtained values direct measurements escort khz resistance measured negative capacitor resistors values according eqs afterwards trimmers negative capacitor finely tuned order achieve decrease transmissibility vibration khz indicated empty circles fig measured transmissibility vibrations fitted theoretical model given eqs following values obtained using method least squares direct measurement using resulted following values respectively noted relative difference fitted measured values resistances varies relative difference much larger allowed relative difference negative capacitor piezoelectric actuator capacitances reason difference presence parasitic capacitances system makes theoretical modeling piezoelectric shunt damping systems difficult adjustment negative capacitors theoretical models practically impossible physics standing behind decrease transmissibility vibrations narrow frequency range easily understood looking fig figures show comparison measured electric impedance absolute value phase piezoelectric actuator calculated values electric impedance absolute value phase negative capacitor three adjustments differ values resistances experiment experiment experiment impedance phase rad transmissibility negative capacitor model narrow model broad model piezoelectric actuator negative capacitor broad band negative capacitor narrow band piezoelectric actuator negative capacitor broad band negative capacitor narrow band impedance abs value real part narrow band broad band imaginary part frequency frequency fig frequency dependences physical quantities control value transmissibility vibrations piezoelectric actuator shunted negative capacitor shown fig shows comparison measured values transmissibility vibrations electrically free piezoelectric actuator filled circles piezoelectric actuator shunted narrow frequency range negative capacitor adjusted khz empty circles broad frequency range negative capacitor adjusted khz empty triangles measured values transmissibility vibrations compared theoretical model absolute value electric impedance piezoelectric actuator measured negative capacitor narrow frequency range see fig broad frequency range see fig reference impedance phase electric impedance piezoelectric actuator measured negative capacitor calculated subtracted calculated real imaginary parts effective spring constant piezoelectric actuator shunted negative capacitor critical frequency figures indicate conditions given eqs satisfied narrow frequency ranges around particular critical frequencies reason narrow frequency ranges decrease real part effective spring constant keff piezoelectric actuator achieved indicated fig next subsection discusses problem broadening frequency range vibration isolation device efficiently suppress vibration transmission transmissibility suppr manual adjustment adaptive adjustment time min fig comparison time dependences vibration isolation efficiency changing operational conditions system manually adjusted negative capacitor solid line system adaptively controlled negative capacitor dashed line respectively vibration isolation system turned time min ambient temperature system changed seen approximately decrease suppression level transmissibility vibrations see minutes system manually adjusted negative capacitor suppression level transmissibility vibrations remains constant adaptive vibration isolation system broad frequency range vibration isolation order broaden frequency range efficiently suppressed vibration transmission necessary achieve precise matching electrical impedances piezoelectric actuator negative capacitor since frequency dependence piezoelectric actuator controlled material construction necessary modify frequency dependence negative capacitor frequency dependence negative capacitor impedance determined reference impedance trivial parallel connection capacitor resistor shown fig replaced complicated network shown fig values capacitances resistances reference impedance adjusted minimize mismatch values frequency range khz khz frequency dependence electric impedance modified reference capacitance measured method least squares yields values fitted values min min min min ikeff fig arg keff ikeff contour plots absolute value argument effective spring constant keff piezoelectric actuator shunted negative capacitor shown fig functions resistances values normalized values min min respectively yield zero absolute value effective spring constant keff keff eff direct measurements using khz giving values trimmers negative capacitor finely tuned order achieve maximum decrease transmissibility vibrations frequency khz transmissibility vibration piezoelectric actuator shunted negative capacitor measured result indicated empty triangles fig seen decrease transmissibility vibration achieved broad frequency range khz khz measured values frequency dependence transmissibility vibrations compared theoretical model given eqs values using method least squares reason broadening frequency range seen fig figures show comparison measured frequency dependence electric impedance absolute value phase piezoelectric actuator calculated values electric impedance negative capacitor narrow broad frequency range reference capacitances shown fig figs seen electric impedances piezoelectric actuator negative capacitor reference capacitance close broad frequency range figure shows frequency dependence real imaginary parts effective young modulus noted decrease young modulus broad frequency range results decrease resonant frequency system approximately yields increase transmissibility vibrations frequencies adaptive system vibration isolation important issue accompanied vibration isolation system manually adjusted negative capacitor shown fig solid line shows time dependence vibration isolation efficiency changed operational conditions system manually adjusted negative capacitor vibration isolation system turned time minute decrease transmissibility vibrations achieved piezoelectric actuator exposed slight heat irradiation bulb tungsten lamp placed distance seen approximately decrease suppression level transmissibility vibrations see minutes avoid severe deteriorative effect changing operational conditions vibration isolation efficiency adaptive vibration isolation system implemented next subsection describes principle control algorithm iterative control law simple control algorithm formulated using analysis contour plots absolute value argument general complex effective spring constant keff piezoelectric actuator shunted negative capacitor negative capacitor shown fig functions resistances plots values normalized values min min respectively shown fig values min min represent optimal values resistances negative capacitor yield zero absolute value effective spring constant one see absolute value keff reaches zero min min interesting graph shown fig argument effective spring constant keff keff ikeff one see value arg keff monotonically increases point goes around optimal adjustment min min direction indicated arrow thus one immediately determine direction optimal adjustment min min respect immediate value measuring argument keff arg keff using principle possible formulate iterative control algorithm follows symbols new old values resistances respectively values resistance increments achievable negative capacitor symbols stand critical values arg keff indicated fig particular values usually determined experimentally next subsection simple way estimation complex value effective spring constant presented estimation effective spring constant effective value spring constant given ratio transmitted force piezoelectric actuator elongation see transmitted force easily measured using piezoelectric force sensor actuator elongation estimated using following idea negative capacitor close required optimal adjustment transmitted force piezoelectric actuator small transmitted force small follows first term much smaller second term situation elongation piezoelectric actuator dominated inverse piezoelectric effect thus proportional voltage applied negative capacitor order estimate argument effective spring constant sufficient calculate phase difference signal force sensor voltage accelerometer mass force sensor negative capacitor force sensor accelerometer accelerometer piezoelectric actuator piezoelectric actuator power amp accelerometer shaker shaker fig combined system measurement transmissibility vibration adaptive vibration isolation system shown side measurement part system consists two accelerometers vibration isolation part system consists piezoelectric actuator shunted electronically adjustable negative capacitor electronic scheme combined system shown side signals accelerometers used calculate transmissibility vibrations signal force sensor applied voltage negative capacitor used estimation effective spring constant keff shunted piezoelectric actuator estimated value argument keff used calculation corrections values resistances electronically adjustable resistors applied negative capacitor arg keff arg arg implementation adaptive vibration isolation system described control algorithm implemented adaptive vibration isolation system shown fig due implementation convenience dataacquisition part adaptive vibration isolation system combined system measurement vibration transmissibility nevertheless two systems independent adaptive vibration isolation system consists force sensor piezoelectric actuator shunted electronically controlled negative capacitor force sensor realized piezoelectric plate charge amplifier kistler arrangement requires calibration done prior experiments setup without damping element transfer function force sensor determined using mass object signal output accelerometer simple fast arrangement allows precise force measurements high frequencies signal force sensor applied voltage negative capacitor used estimation effective spring constant keff shunted piezoelectric actuator estimated value argument keff used calculation corrections values resistances electronically adjustable resistors according eqs order make electronic control resistances negative capacitor possible manually adjusted trimmers replaced electronically controlled resistors implemented pair diode photoresistor example measured characteristics electronically adjustable resistor shown fig voltage controls current diode therefore intensity emitted light using converter intensity generated light controls resistance photoresistor instantaneous values incident transmitted vibrations measured piezoelectric accelerometers accelerometers resonant frequency khz ensures flat phase correct transmission function frequency range experiments signals accelerometers amplified icp amplifier electric signals accelerometers force sensor electric voltage applied piezoelectric actuator negative capacitor measured digitized data acquisition card stressed accelerometers part measurement system used measure transmissibility vibrations evaluate efficiency adaptive vibration isolation system accelerometers used control vibration transmission signals accelerometers enter negative capacitor shunt used iterative control law personal computer used three independent simultaneous operations first used generation signal incident vibrations matlab software signal dominant harmonic components generated output signal introduced amplifier fed piezoelectric shaker second processes signals accelerometers calculates frequency dependence transmissibility vibrations third processes signals voltage controlled resistor resistance control voltage fig example measured characteristics electronically adjustable resistor constructed pair diode photoresistor voltage controls current diode therefore intensity emitted light using converter intensity generated light controls resistance photoresistor force sensor negative capacitor generates control signals electronically adjustable resistors negative capacitor according iterative control law transmissibility vibrations harmonic time dependence frequency khz adaptive vibration isolation system shown using dashed line fig seen transmissibility vibrations remained constant even varying operational conditions ambient temperature however noted situation vibrations harmonic time dependence estimation effective spring constant argument straightforward easy task hand consideration harmonic vibrations greatly limits applicability vibration isolation device order eliminate drawback broaden applicability described adaptive vibration isolation system modification control algorithm necessary described next subsection suppression vibrations general time dependence real situations incident vibrations usually consist sum several randomly changing dominant harmonic components harmonic components appear system due mechanical parts due vibration revolving ical parts order suppress vibration transmission vibrating mechanical parts real industrial applications following modification control algorithm implemented first signals force sensor voltage applied negative capacitor measured amplitude signal force sensor exceeds arbitrarily chosen threshold fast fourier transformation applied time dependencies measured signals order obtain amplitude phase frequency spectra distribution vibration power along frequency axis analyzed dominant harmonic component greatest amplitude found dominant harmonic component selected suppressed selected frequency dominant harmonic component phase difference dominant harmonic components signals force sensor negative capacitor output calculated calculated value phase difference used iterative corrections values resistances according eqs application corrections values resistances new signals force sensor negative capacitor output measured steps periodically repeated dominant frequency suppressed force sensor signal measurable level order evaluate performance adaptive broad band vibration suppression device five different vibration signals generated applied vibration isolation device vibration signal consists random noise one dominant harmonic component given frequency figure shows spectra five force signals transmitted vibration isolation system different frequencies dominant harmonic component solid black line indicates force amplitude spectra transmitted piezoelectric actuator disconnected negative capacitor solid blue lines indicate amplitude spectra force transmitted piezoelectric actuator shunted negative capacitor frequency dependences transmissibility vibrations piezoelectric actuator shunted adaptive broad frequency range negative capacitor five aforementioned vibration signals shown fig seen adaptive control algorithm adjusts negative capacitor way transmissibility vibration curve minimum around frequency dominant harmonic component vibration signal figure also indicates shifts mechanical dominant harmonic component frequency khz dominant harmonic component frequency khz force amplitude dominant harmonic component frequency khz dominant harmonic component frequency khz dominant harmonic component frequency khz frequency fig spectra five force signals transmitted vibration isolation system different frequencies dominant harmonic component solid black line indicates force amplitude spectra transmitted piezoelectric actuator disconnected negative capacitor solid blue lines indicates amplitude spectra force transmitted piezoelectric actuator shunted negative capacitor vibration signal consists random noise one dominant harmonic component given frequency resonant frequency system due reduction effective spring constant piezoelectric actuator using negative capacitor broad frequency range yields increase transmissibility vibrations subresonant frequencies unwanted phenomenon easily eliminated inserting filter negative capacitor negative capacitor negative capacitor khz khz khz khz khz transmissibility frequency fig frequency dependences transmissibility vibrations piezoelectric actuator shunted adaptive negative capacitor curve corresponds transfer functions adaptive system adjusted cancel vibration signal force amplitude spectra shown fig noted decrease young modulus broad frequency range results decrease resonant frequency system yields increase transmissibility vibrations subresonant frequencies finally noted iterative control algorithm conveniently compensate effects dielectric nonlinearity piezoelectric actuator point increase amplitude incident vibrations voltage applied operational amplifier also increased less proportionally increase incident vibration amplitude permittivity capacitance piezoelectric actuator seen negative capacitor slightly changed due dielectric nonlinearity usually according rayleigh law causes maladjustment system unwanted drop efficiency vibration isolation however change amplitude incident vibration extremely fast system remains stable iterative control algorithm quickly compensates changes piezoelectric actuator capacitance behavior expected even full voltage range actuator achieved systems standard amplifier presented work fleming moheimani conclusions theoretical model vibration transmission piezoelectric actuator shunted negative capacitor presented model verified using experiments performed narrow frequency pure tone vibration isolation proper modification reference capacitor negative capacitor successfully demonstrated possible achieve vibration transmission suppression broad frequency range khz khz iterative control law automatic adjustment negative capacitor disclosed using analyzing absolute value argument effective spring constant piezoelectric actuator shunted negative capacitor method estimation effective spring constant argument vibration isolation system presented however adaptive system basic arrangement applicable suppression vibrations harmonic time dependences order eliminate drawback evolved signal processing implemented shown iterative control algorithm applicable also vibrations general time dependence advantages presented system suppression vibration transmission stem simple electronic realization using analog circuit operational amplifier broad frequency range efficiently suppressed vibrations khz khz simple control law allows applying automatic corrections negative capacitor system work varying operating conditions addition presented adaptive system example general concept adaptive piezoelectric shunt damping easily modified applied variety different types piezoelectric actuators electroacoustic transducers presented realization vibration isolation device offers solution many real noise vibration problems aknowledgments work supported czech science foundation project gacr student grant sgs interactive mechatronics systems using cybernetics principles european regional development fund ministry education youth sports czech republic project research center special optics optoelectronic systems toptec authors acknowledge julie reading manuscript hagood von flotow damping structural vibrations piezoelectric materials passive electrical networks journal sound vibration vol apr moheimani fleming piezoelectric transducers vibration control damping tsai wang structural damping characteristics active piezoelectric actuators passive shunt journal sound vibration vol petit lefeuvre richard guyomar broadband semi passive piezoelectric technique structural damping smart structures materials damping isolation wang vol proceedings society instrumentation engineers spie morgan wang piezoelectric absorbers systems multiple harmonic excitations journal sound vibration vol morgan wang piezoelectric absorber structural vibration control harmonic excitations frequency part algorithm development analysis journal vibration asme vol jan ming yuan hongli active control sound transmission stiffened panel using hybrid control strategy journal intelligent material systems structures vol behrens fleming moheimani broadband controller shunt piezoelectric damping structural vibration smart materials structures vol feb niederberger fleming moheimani morari adaptive resonant piezoelectric shunt damping smart materials structures vol oct fleming moheimani adaptive piezoelectric shunt damping smart materials structures vol feb badel sebald guyomar lallart lefeuvre richard qiu piezoelectric vibration control synchronized switching adaptive voltage sources towards wideband damping journal acoustical society america vol may date kutani sakai electrically controlled elasticity utilizing piezoelectric coupling journal applied physics vol nic fukada yamamoto noise shielding system utilizing thin piezoelectric membrane elasticity control journal applied physics vol fukada yamamoto sound absorbing system application active elasticity control technique journal applied physics vol imoto nishiura yamamoto date fukada tajitsu elasticity control piezoelectric lead zirconate titanate pzt materials using circuits japanese journal applied physics vol tahara ueda takarada imoto yamamoto date fukada tajitsu basic study application elasticity control piezoelectric lead zirconate titanate materials using circuits sound shielding technology japanese journal applied physics vol kodama date yamamoto fukada study sound shielding control curved piezoelectric sheets connected negative capacitance circuits journal sound vibration vol apr sluka feedback control piezoelectric actuator elastic properties vibration isolation system ferroelectrics vol european conference applications polar dielectrics metz france sep preumont marneffe deraemaeker bossens damping truss structure piezoelectric transducer computers structures vol feb eccomas thematic conference smart structures materials lisbon portugal jul sluka kodama fukada sound shielding piezoelectric membrane negative capacitor feedback control ieee transactions ultrasonics ferroelectrics frequency control vol aug fleming moheimani improved current charge amplifiers driving piezoelectric loads issues signal processing design synthesis shunt damping circuits journal intelligent material systems structures vol
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transfer learning deep learning data jaekoo lee hyunjae kim jongsun lee sungroh yoon dec electrical computer engineering seoul national university seoul republic korea sryoon abstract graphs provide powerful means representing complex interactions entities recently new deep learning approaches emerged representing modeling graphstructured data conventional deep learning methods convolutional neural networks recurrent neural networks mainly focused inputs image audio leveraged representation learning capabilities deep techniques detect structural characteristics graphs giving promising results graph applications paper attempt advance deep learning data incorporating another component transfer learning transferring intrinsic geometric information learned source domain approach construct model new related task target domain without collecting new data without training new model scratch thoroughly tested approach text data confirmed effectiveness proposed transfer learning framework deep learning graphs according experiments transfer learning effective source target domains bear high level structural similarity graph representations introduction recently many deep neural network models adopted successfully various fields lecun bengio hinton schmidhuber particular convolutional neural networks cnn krizhevsky sutskever hinton image video recognition recurrent neural networks rnn sutskever vinyals speech natural language processing nlp often deliver unprecedented levels performance deep learning also triggered advances implementing intelligence game silver cnn rnn extract features input data image video audio data structured typically regular grids see fig top grid structures often assumed statistical characteristics stationarity locality facilitate modeling process learning algorithms take advantage assumption boost performance recopyright association advancement artificial intelligence rights reserved ducing complexity parameters schmidhuber bruna henaff bruna lecun reality exist wide variety data types need general structures represent model complex interactions among entities examples include social media mining protein interaction studies applications graph provide natural way representing entities interactions deo input challenging find statistical characteristics assumed gridstructured input bruna henaff bruna lecun theoretical challenges including practical limitations data training efficiency make difficult apply conventional deep learning approaches directly igniting research adapting deep learning data bruna henaff bruna lecun jain many graph analysis methods structural properties derived input graphs play crucial role uncovering hidden patterns koutra vogelstein faloutsos lee kim yoon representation learning capability deep networks useful automatically detecting structural features deep learning approaches reported promising results paper attempt advance deep learning data incorporating another key component transfer learning pan yang overcoming common assumption training test data drawn feature space distribution transfer learning task domains alleviate burden collecting data training models new task given importance structural characteristics graph analysis core proposal transfer structural features learned deep networks source domain target domain informally shown fig bottom context graphs call transferred information intrinsic geometric information starting intuitive baseline need fill many details implement transfer learning deep learning graph data particular need answer two important questions condition expect successful knowledge transfer task domains actually perform transfer domain input hidden layers graph structure space grid structure space convolution output fully connected pooling convolution graphs pooling fully connected graph spectrum transform graph laplacian transfer learning intrinsic geometric information figure conventional cnn works regular grid domain top proposed transfer learning framework cnn transfer intrinsic geometric information obtained source graph domain target graph domain bottom tively paper tries address questions demonstrate effectiveness approach tested public nlp datasets text classification zhang zhao lecun dataset contained corpus news articles internet reviews ontology entries represented dataset amazon reviews graph capture interactions among words dataset used spectral cnn scnn bruna henaff bruna lecun model graph using neural networks learned model used classifying unseen texts data source amazon furthermore experimental results confirmed transfer learning methodology allows implicitly derive model classifying texts another source yelp reviews without collecting new data without repeating learning procedures scratch specific contributions summarized follows proposed new transfer learning framework deep learning input data structure graphs best authors knowledge work first attempt kind adopting approach relieve burden data models related tasks address investigated conditions successful knowledge transfers graph domains conjectured two graphs similar structural characteristics would give better results confirmed comparing graph similarity transfer learning accuracy answer tested diverse alternatives components proposed framework graph generation input representation deep network construction particular improve scnn model extracting datadriven structural features graphs analyzed optimized key factors affect performance scnn method quantify spectral features graph performed extensive set experiments using synthetic data show effectiveness approach related work graphs provide general way representing diverse interactions entities studied extensively sonawane kulkarni addition studies representation quantification relations similarities koutra vogelstein faloutsos various studies focused graph data use structural information recently deep learning methods automatically extract structural characteristics graphs proposed duvenaud examples deep learning applied noneuclidean space include graph wavelets applying deep graphs using properties automatically extracted features rustamov guibas analysis molecular fingerprints proteins saved graphs duvenaud model handling tree structures context programming language processing mou particularly relevant approach localized scnn model boscaini deep learning approach extract properties deformable shapes generalized scnn model bruna henaff bruna lecun key component framework borrowed fourier transform concept signal processing field order apply cnns grid domain domain model convolutional operation graphs recently restricted boltzmann machine lecun bengio hinton used learn structural features graphs unsupervised manner classification niepert ahmed kutzkov efficient scalable classification data approximation spectral graph convolutions utilized kipf welling method could adopt approaches base learning model improve effectiveness transfer learning proposed method fig presents diagram illustrating overall flow proposed method consists five steps first three steps produce graph input identify unique structural features graph last two steps apply transfer learning based learned features graph similarity carry inference step graph production represent data elements input data interactions relations nodes edges respectively graph input dataset construct undirected connected weighted graph represent sets vertices edges respectively denotes weighted adjacency matrix assume utilize two recent techniques derive graph specifically edge set input data graph estimation coge sonawane kulkarni supervised graph estimation sge henaff bruna lecun coge directly quantifies closeness data elements based frequency sge automatically learns similarity features among elements fully connected network model representation graphs spectral domain extract intrinsic geometric characteristics entire graph deriving normalized laplacian matrix graph constructed step graph domain provides values graph spectral bases convolution operation scnn mohar koutra vogelstein faloutsos consider three types laplacian lbasic random normalized laplacian lrw random walk restart based normalized laplacian lrwr given tong faloutsos pan lbasic lrw rwr represents degree graph represents probability restart note approximation attained attenuating neighboring influence approximation attained belief propagation fast approximation symmetric matrix decomposed diagonalization combining eigenvalues diagonal matrix shows degree number edges attached node corresponding orthogonal eigenvectors order eigenvalue index node mohar recall function defined nodes graph represented vector dimension indicating value vertex shuman shuman ricaud vandergheynst fourier transform eigenfunctions represent function defined nodes graph transformed function represented set basis eigenvectors parseval theorem also holds energy transformed function original function two functions verifying consistency two domains chung shuman ricaud vandergheynst indicates input function defined vertex domain graph converted corresponding graph spectral domain using concept fourier analysis graphs generalized convolutional operation denoted functions defined diagonalized linear multiplication spectral domain follows bruna henaff bruna lecun shuman ricaud vandergheynst also expressed matrix eigenvectors graph laplacian columns quantify intrinsic structural geometry entire graph domain serve spectral bases graph matrix functions fourier transform graph spectral domain regard receptive filter learned convolution operation convolution layer cnn regular grid domain regarded matrix diagonalized elements input defined graph domain provided conventional cnns output layer defined number feature maps input layer nonlinear function filter matrix feature map feature map scnn transform input size coge input source domain data set coge input target domain data set layers model output fully connected pooling learning transferable features convolution graph graph laplacian applying convolution networks graph producing graph solid line training process dotted line transfer learning process representation graphs spectral domain transfer learning spectral domain figure overview proposed method output size given nonlinear function diagonal matrix implies training weights learnable filters training multipliers eigenvalues laplacian bruna henaff bruna lecun characterizes scnn generalized cnn model several filter banks generalized convolutional operations graph augment scnn model support spatial locality made independent input size using windowed smoothing filters defined based polynomial kernel degree shuman ricaud vandergheynst based fact originally observed signal processing smoothness spectral domain spatial decay local features original domain implement idea using eigenvectors subsampled laplacian eigenvectors laplacian boscaini applying convolutional networks graphs train scnn model using information obtained previous steps represent geometric information local behaviors surface structural graph domain model hierarchical structure consisting layers convolutional pooling fully connected layer shown fig training determines weights layer minimizing cost loss function model learn various features convolution operation spectral information structural graph domain bruna henaff bruna lecun learning transferable features model training completed contains datadriven features data derived input steps stated introduction core proposal transfer information structural characteristics graph learned deep learning features learned step provide information transfer learning spectral domain according pan yang domain context transfer learning consists feature space probability distribution given domain denote task label space predictive function learned training data objective general transfer learning improve learning target domain exploiting knowledge source domain task present context transfer intrinsic geometric information learned graph encoding knowledge steps skip steps generate well steps extract structural characteristics therefrom condition bear structural similarities directly build model copying convolutional pooling layers contain features trained training fully connected layer model fine tuning weights way transfer learning provides efficiency learning also helps minimize problems resulting lack data imperfect structural information new task note proposed method guarantees spectral homogeneity graphs using union node sets heterogeneous source target datasets method possible utilize spectral features graphs heterogeneous datasets table details datasets used train test class name dbp yelp amaz dbp yelp amaz sim corr corpus news articles web dbp ontology data dbpedia yelp reviews yelp amaz reviews amazon sim indicates two graphs structurally complementary whereas value means identical corr represents correlation bag words extracted text corpora results discussion tested proposed method performing topic classification text documents text data carry information individual words also relationships methods widely used text mining utilized public nlp data zhang zhao lecun contained multiple corpora news articles internet reviews ontology entries table controlled experiments also generated two pairs synthetic datasets random sampling real corpora one pair consisted two corpora high similarity pair consisted two corpora low similarity measuring structural similarity graphs shown table fig used methods reported existing studies koutra vogelstein faloutsos lee kim yoon refer note table details note yelp amaz bear highest similarity terms metrics used implemented deep networks torch cuda using adagrad optimizer relu activation carried cross validation note proposed method offer efficient training scheme relatively low computation cost leaving eigenvalue decomposition scnn model trained data source domain experiments proposed method provided reduction average training time using setting first carried comprehensive experiments determine factors affected performance scnn model graph modeling table lists part results obtained varying net architecture method generate graphs type laplacian matrix along resulting classification accuracy combination observe table laplacian methods significantly affect performance lrwr benefit terms computational complexity sge tended give accurate results coge implies initial graph generation affected model training critically structural table performance scnn model various hyperparameters text topic classification task model architecture graph gener type coge coge coge sge sge sge coge coge coge sge sge sge coge coge coge sge sge sge lbasic lrw lrwr lbasic lrw lrwr lbasic lrw lrwr lbasic lrw lrwr lbasic lrw lrwr lbasic lrw lrwr classification accuracy dbp yelp amaz training set kernel degree learning rate used cost function adagrad optimizer means use graph convolutional layers feature maps means use fully connected layer hidden units feature extraction experiments shown table model refer note table notation gave best results used model main learning model following experiments performed experiments determine effectiveness transfer learning using synthetic datasets results shown fig plots top row pair synthetic corpora high similarity sim corr varying quantities data training transferred model target domain entire target data plots bottom row fig correspond results pair synthetic corpora low similarity sim corr observe transfer learning effective higher similarity case test accuracy transferred model increased significantly faster source domain model using target domain data sufficient training using data provide noticeable difference lower similarity case training target domain limited could deliver level accuracy source domain due discrepancies underlying structure source target domains finally tested approach four corpora dbp yelp amaz shown fig plots top row represent test accuracy model trained original data solid line transferred model trained data dotted line bottom plots represent test loss two corpora highest level similarity yelp amaz effect percentage target dataset used training transferred model source target test accuracy sim corr sim corr source target epochs figure results intrinsic geometric information transfer learning synthetic datasets best viewed color top source target datasets high similarity graph representations bottom source target datasets low similarity column percentage target dataset used training transferred model fine tuning fully connected layer repeated every experiment times data point shows boxplot red source domain blue target domain lines connect median locations boxplots transfer learning salient test accuracy transferred model comparable source model yelp lower amaz cases lower similarity yelp amaz transfer learning less effective results confirmed observation knowledge transfer successful source target domains high level structural similarity underlying graph representations conclusion proposed new transfer learning framework deep learning data approach transfer intrinsic geometric information learned graph representation source domain target domain observed knowledge transfer tasks domains effective source target domains possess high similarity graph representations anticipate adoption methodology help extend territory deep learning data structure well cases limited quantity quality data prove planning apply approach diverse datasets different domains acknowledgments work supported ministry science ict future planning institute information communications technology promotion brain korea plus project grant funded korea government samsung research funding center samsung electronics references boscaini boscaini masci melzi bronstein castellani vandergheynst learning descriptors deformable shapes using localized spectral convolutional networks computer graphics forum volume wiley online library bruna bruna zaremba szlam lecun spectral networks locally connected networks graphs arxiv preprint chung chung spectral graph theory volume american mathematical soc deo deo graph theory applications engineering computer science courier dover publications duvenaud duvenaud maclaurin iparraguirre bombarell hirzel adams convolutional networks graphs learning molecular fingerprints advances neural information processing systems henaff bruna lecun henaff bruna lecun deep convolutional networks graphstructured data arxiv preprint jain jain zamir savarese saxena deep learning spatiotemporal graphs arxiv preprint dbp yelp amaz test loss test accuracy target domain epochs figure results proposed method datasets best viewed color plot top row test accuracy model trained original data solid lines models trained data sources transferred dotted lines plot bottom row test loss yelp amaz transfer learning effective given highest level structural similarity cases see table kipf welling kipf welling classification graph convolutional networks arxiv preprint koutra vogelstein faloutsos koutra vogelstein faloutsos deltacon principled similarity function siam krizhevsky sutskever hinton krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems lecun bengio hinton lecun bengio hinton deep learning nature lee kim yoon lee kim yoon measuring dynamic graph similarity ricom rwr intergraph compression ieee international conference data mining tarlow brockschmidt zemel gated graph sequence neural networks arxiv preprint mohar mohar applications laplace eigenvalues graphs springer mou mou zhang wang jin convolutional neural networks tree structures programming language processing aaai conference artificial intelligence niepert ahmed kutzkov niepert ahmed kutzkov learning convolutional neural networks graphs arxiv preprint pan yang pan yang survey transfer learning ieee transactions knowledge data engineering rustamov guibas rustamov guibas wavelets graphs via deep learning advances neural information processing systems schmidhuber schmidhuber deep learning neural networks overview neural networks shuman shuman narang frossard ortega vandergheynst emerging field signal processing graphs extending highdimensional data analysis networks irregular domains ieee signal processing magazine shuman ricaud vandergheynst shuman ricaud vandergheynst analysis graphs applied computational harmonic analysis silver silver huang maddison guez sifre van den driessche schrittwieser antonoglou panneershelvam lanctot mastering game deep neural networks tree search nature sonawane kulkarni sonawane kulkarni graph based representation analysis text document survey techniques international journal computer applications sutskever vinyals sutskever vinyals sequence sequence learning neural networks advances neural information processing systems tong faloutsos pan tong faloutsos pan fast random walk restart applications ieee international conference data mining zhang zhao lecun zhang zhao lecun convolutional networks text classification advances neural information processing systems
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jan general approach cure models survival analysis valentin patilea ingrid van keilegom january abstract survival analysis often happens subjects study experience event interest considered cured population thus mixture two subpopulations one cured subjects one susceptible subjects covariates present mixture cure model used model conditional survival function population depends two components probability cured conditional survival function susceptible subjects paper propose novel approach estimate mixture cure model data subject random right censoring work parametric model cure proportion like logistic model conditional survival function uncured subjects unspecified approach based inversion allows write survival function function distribution observable random variables leads general class models allows flexible rich modeling conditional survival function show identifiability proposed model well weak consistency asymptotic normality model parameters also consider detail case kernel estimators used nonparametric part model new estimators compared estimators cox mixture cure model via finite sample simulations finally apply new model estimation procedure two medical data sets key words asymptotic normality bootstrap kernel smoothing logistic regression mixture cure model semiparametric model crest ensai france patilea acknowledges support research program new challenges new data lcl genes email address patilea orstat katholieke universiteit leuven belgium van keilegom acknowledges support european research council horizon erc grant agreement iap research network belgian state email address introduction driven emerging applications last two decades increasing interest analysis models allowing situation fraction right censored observed lifetimes corresponds subjects never experience event biostatistics models including covariates usually called cure models allow positive cure fraction corresponds proportion patients cured disease review models survival analysis see instance maller zhou peng taylor economists sometimes call models split population models see schmidt witte reliability engineers refer limitedfailure population life models meeker first sight cure regression model nothing binary outcome cured versus uncured regression problem difficulty comes fact cured subjects unlabeled observations among censored data one use observations censored uncensored complete missing information thus identify estimate make inference cure fraction regression function propose general approach task tool provides general ground cure regression models idea start laws observed variables express quantities interest cure rate conditional survival uncured subjects functionals laws general expressions call inversion formulae derive particular constraint space covariates vehicles allow wide modeling choice parametric semiparametric nonparametric law lifetime interest cure rate indeed inversion formulae allow express likelihood binary outcome model function laws observed variables likelihood estimator parameter vector cure fraction function simply maximizer likelihood obtained replacing laws observations estimators hand estimate parameter cure fraction inversion formulae provide estimate conditional survival uncured subjects sake clarity focus mixture cure models parametric cure fraction function type model popular among practitioners meanwhile lifetime interest left unspecified paper organized follows section provide general description mixture cure models next introduce needed notation present inversion formulae approach built finish section discussion identification issue new insight existing approaches literature cure models section introduces general maximum likelihood estimator section derive general asymptotic results simple bootstrap procedure making feasible inference proposed section ends illustration general approach case conditional law observations estimated kernel smoothing sections report empirical results obtained simulated two real data sets estimator performs well simulations provides similar interpretable results applications compared competing hazards mixture approach technical proofs relegated appendix model general class mixture cure models let denote possible monotone transformation lifetime interest takes values cured observation corresponds event following event allowed positive probability let covariate vector support belonging general covariate space covariate vector could include discrete continuous components survival function written depending model used one obtains parametric semiparametric nonparametric model called mixture cure model literature one often assumes follows logistic model exp exp recently semiparametric models like model amico nonparametric models peng proposed survival function susceptible subjects variety models proposed including parametric models see boag farewell semiparametric models based proportional hazards assumption see kuk chen taylor fang see also othus nonparametric models see taylor peng paper propose model parametrically assume belongs family conditional probability functions takes values interval parameter vector model parameter set family could logistic family parametric family survival function impose assumptions order flexible rich class models choose later see estimation estimator satisfies certain minimal conditions used hence allow large variety parametric semiparametric nonparametric estimation methods often case data assume lifetime subject random right censoring instead observing observe pair random variable called censoring time identification assumptions required able identify conditional law observed variables let assume conditional independence usual identification assumption survival analysis presence covariates zero probability infinity condition implies almost surely latter mild condition required admit observations finite case common applications sake simplicity let also consider condition commonly used survival analysis implies notations preliminaries start preliminary arguments valid general without assuming model functions observations characterized conditional def since assume finite let sup denote right endpoint support conditional let define similar way note max note equal infinity even though takes finite values let define conditional probabilities let show probability cured could identified observations without reference model probability conditions write equations could solved thus allow express functions unique way explicit transformations functions purpose let consider conditional cumulative hazard measures model equations yield write following functionals stands set see gill johansen moreover way identify conditional law beyond therefore impose note condition longer identification restriction simple consequence definition finally condition saying assume let point condition satisfied indeed necessarily hence thus contradicts important understand two conditional satisfying conditions define uniquely indeed precisely probability distribution given mass beyond concentrated infinity general functionals assume conditions throughout paper key point new approach inversion formulae write thus consider conditional cumulative hazard measure finite values lifetime interest def since using relationship obtain next using write let recall written transformation see equations representation surprising since consider lifetime interest hence plays role censoring variable hence estimating conditional distribution function complicated classical conditional setup since fact could equal infinity positive conditional probability irrelevant estimating finally representation given equation plugged equation allows express thus maps measures key element providing insight existing approaches starting point new approach model identification issues let investigate identification issue recall model involves functions assumptions fixed value parameter let let denote conditional law given moreover let equations define conditional law observations based model precisely choice model yields conditional law given model correctly specified exists value remaining question whether true value parameter identifiable words one check given conditional subdistributions exists unique satisfying condition purpose impose following mild condition show almost surely indeed condition guarantees necessarily positive thus could simplified last display deduce finally recall construction model consider coincides conditional law given thus taking integrals sides last let gather facts following statement display obtain theorem conditions model identifiable interpreting previous modeling approaches suppose function follows logistic model comment several models considered literature parametric proportional hazards mixture model parametric modeling one usually supposes belongs parametric family cumulative hazard functions like instance weibull model see farewell several contributions proposed flexible semiparametric proportional hazards approach see fang references therein model one imposes structure measure precisely supposed exp parameter estimated unknown baseline cumulative hazard function inversion formulae reveal approach parameters depend observed conditional measures also parameter true parametric models mixture cure model taylor suggested estimate using type estimator approach one implicitly assumes law given given depend equivalent supposing next estimate one modify unconditional version usual inversion formulae take account conditional probability event following taylor approach rewrite next assume last equality remains true replaced unconditional versions assume see equations taylor equation could solved iteratively procedure given iteration build updated estimate see taylor details let point even independent thus depend subdistribution still depends since hence natural form equation investigation procedure based latter equation considered elsewhere maximum likelihood estimation let sample copies vector use likelihood approach based formulae build estimator build likelihood use estimates butions estimates constructed sample without reference model conditional probability stage derive asymptotic results necessary impose particular form impose estimators satisfy mild conditions let instead defined equations fbc estimator obtained equations fbc let denote density covariate vector respect dominating measure contribution observation likelihood fbc contribution fbc since laws censoring variable covariate vector carry information parameter drop factors fbc fbc hence criterion maximized respect estimator propose arg max log let review identification issue context likelihood estimation approach conditions hold true parametric model conditional probability event correct identifiable sense condition true parameter value identified condition conclusion theorem remains check proposed likelihood approach allows consistently estimate let log log log following common notational convention see instance gill treat length small time interval also name interval moreover use convention let notice additive terms containing function log expected limit hence minimal condition guaranteeing consistency random function log likelihood estimation approach maximizer limit likelihood criterion log proved following proposition using bernoulli sample likelihood inequality proof given appendix proposition suppose conditions hold true value parameter defined equation log log general asymptotic results little assumptions needed analysis far proceed asymptotic results need specific respect several aspects order prove along sequences values consistency control asymptotic behavior parameter control requires control denominators like fbc support uniformly respect usual way deal technical difficulty consider finite threshold beyond uncensored lifetime observed inf moreover able keep denominators away zero require condition inf particular condition implies moreover given condition necessarily means constraint could relaxed expense suitable adjustments inversion formulae simplicity keep condition let also notice condition implies inf inf conditions like equations less explicitly used literature cure models sometimes justified representing total study instance supposes min min min conditional probability cured precisely conditional probability event next supposes inf cumulative hazard function conditions together clearly imply conditions fang implicitly restrict uncensored lifetimes compact interval suppose could possible set values positive probability proportional hazards context covariates taking values bounded set assumed fang equivalent almost constant fact technical conditions similar conditions could traced cure models literature unexpected view section indeed existing approaches could interpreted inversion formulae thus technical problems face asymptotic investigation expected also present alternative approaches consistency let sketch arguments use proof theorem deriving one hand conditional subdistributions given one consistency build purely parametric likelihood construction functional functional estimated versions hence prerequisite condition deriving consistency semiparametric estimator consistency infeasible maximum likelihood estimator arg max log necessary condition consistency log sup log log log derive consistency using section van der vaart see proof appendix details prove condition guarantee uniform convergence stated assumption indeed uniform convergence imply sup sup sup sup sup see lemma appendix uniform convergence equation follows need following assumptions prove consistency appearing conditions sup sup parameter set compact exist constants inf inf state consistency result theorem assume hold true moreover assume exists unique value parameter space true let point consistency result stated terms subdistributions observations conditional probability model identification assumptions used proposition hold true model correctly specified consistently estimates cure probability support let also notice condition guarantees property certain classes functions could significantly weakened applications condition cover common modeling situations condition weak condition model satisfied logistic model compact asymptotic normality asymptotic normality use approach chen respect purpose use derivative log respect first note vector partial derivatives log components equals log log log log log fbt log defined proof theorem embed nuisance functions develop asymptotic normality estimator functional space equipped space chosen depending estimators satisfy certain conditions give true vector nuisance functions let measures associated functions define log note hence kmn kmn addition thus map decreasing moreover condition inf need lower bound valid neighborhood around hence let consider neighborhood inf inf existence guaranteed condition regularity function see assumption strengthens assumption finally let note construction thus arguments guaranteeing existence set equation inf inf inf define derivative direction lim similar way derivatives defined need following assumptions matrix exists neighborhood continuous moreover function continuously differentiable derivative bounded uniformly moreover compact belongs interior satisfies following estimator iii exist functions denotes conditional expectation given data functions defined appendix note expectations conditionally sample taken respect generic variables law sample log class satisfies number space respect norm smallest number balls needed cover space theorem assume hold true var bootstrap consistency although principle one use theorem making inference asymptotic variance complicated structure estimation would cumbersome precision small samples could moreover rather poor continue section showing bootstrap procedure used estimate asymptotic approximate whole distribution construct confidence intervals variance test hypotheses regarding propose use naive bootstrap procedure consisting drawing triplets randomly replacement data estimator based bootstrap data let let define bootstrap estimator sequence satisfies following result shows bootstrap works sense allows recover correctly distribution theorem assume hold true moreover assume continuous respect replaced holds true sup denotes probability conditionally data inequality sign means inequality vectors verification assumptions kernel estimators finish section illustration verification assumptions asymptotic results conditional subdistributions estimated means kernel smoothing consider case composed continuous discrete components rdc rdd simplicity assume support discrete subvector finite also assume life time transformed logarithmic transformation support subdistributions could estimated means kernel estimator khn khn bandwidth sequence udc probability density function nonparametric smoothing continuous covariates possible dimensions larger however technical arguments necessary verify assumptions used asymptotic results tedious therefore following consider discrete covariates contribute curse dimensionality therefore could larger however simplicity consider discrete covariates satisfy assumption need impose following conditions sequence satisfies log support compact subset probability density function compact support twice continuously differentiable let space functions variation bounded let space continuously differentiable functions satisfy let define following norm associated space let sup sup follows propositions akritas van keilegom provided condition moreover log supx nhn log see proposition akritas van keilegom class satisfies assumption thanks lemma lopez remains show validity assumption iii show first statement second one shown similar way note left hand side equals dhk dhk required form simulations section investigate small sample performance estimation method consider following model covariate generated uniform distribution conditional probability cured follows logistic model exp exp work corresponding average cure rate respectively conditional distribution function uncured individuals constructed follows given draw exponential distribution mean equal exp next order respect condition truncate distribution quantile order exponential distribution mean exp exp exp note distribution function corresponding cox model baseline hazard equal exponential factor equal exp next generate censoring variable independently exponential distribution mean equal mean way respectively censoring follows compare estimator estimator proposed assumes cox model uncured individuals exponential factor cox model assumed linear covariate hence cox model verified estimated coefficients cox model obtained using package smcure estimation procedure used kernel estimators given section programmed using optimization procedure optim starting values used estimator obtained logistic model based censoring indicator surrogate unobserved cure indicator however due likelihood function due inconsistency vector starting values procedure optim often ends local maximum instead global maximum circumvent problem added following intermediate step estimation procedure based initial starting values estimate logistic model based nonparametric estimator fbt maximize fbt log fbt log since concave unique local global maximum expected close maximizer likelihood use intermediate estimate starting value likelihood maximization results maximization procedure given table case table case total samples size generated tables show bias mean squared error mse estimators obtained cox model procedure kernel function taken equal epanechnikov kernel bandwidth kernel estimators taken proportional verify regularity condition several values namely addition also used procedure proposed lin racine kernel estimators conditional distribution functions procedure implemented package used average sample simulation calculated bandwidths two bandwidths tables show estimator outperforms one based cox model even cox model correct also show estimator mildly sensitive bandwidth could explained fact average effect bandwidth also see selection bandwidth working rather well sense mse close smallest value among mse corresponding three fixed bandwidths next look estimation quartiles distribution estimate quartiles means nonparametric estimator means cox model studied results given tables show could expected cox model satisfied par bias mse bias mse bias mse hcv bias mse cox bias mse table bias mse two sample sizes three values three bandwidths form bandwidth hcv obtained cured censoring cox model satisfied par bias mse bias mse bias mse hcv bias mse cox bias mse table bias mse two sample sizes three values three bandwidths form bandwidth hcv obtained cured censoring cox model satisfied mse quartiles obtained cox model much higher corresponding mse obtained procedure shows importance model bias mse bias mse bias mse hcv bias mse cox bias mse table bias mse conditional quantiles order two sample sizes three bandwidths form bandwidth hcv obtained cured censoring cox model satisfied impose assumptions distribution uncured individuals still provides accurate estimators logistic part model also verify close distributions normal distribution know thanks theorem estimators converge normal limit tends infinity figure shows distribution rather close normal limit especially figure based samples generated model cured censoring results shown space constraints close straight line showing results improve increases finally verify accuracy naive bootstrap proposed section consider model restrict attention case cured censoring figure shows boxplots variance obtained bootstrap resamples samples bandwidth empirical variance estimators also added shows bootstrap variance well centered around corresponding empirical variance bias mse bias mse bias mse hcv bias mse cox bias mse table bias mse conditional quantiles order two sample sizes three bandwidths form bandwidth hcv obtained cured censoring cox model satisfied data analysis let apply estimation procedure two medical data sets first one breast cancer patients breast cancer treated wang event interest associated survival time distant survival time defined time first distant progression death whichever comes first patients experience relapse breast cancer plot estimator data given figure shows large plateau furthermore large proportion censored observations plateau suggests cure model appropriate data covariate use age patients ranges years average age years estimate using estimator using estimator based cox model bandwidth selected using simulation section estimated theoretical quantiles theoretical quantiles theoretical quantiles theoretical quantiles sample quantiles sample quantiles sample quantiles theoretical quantiles sample quantiles sample quantiles sample quantiles theoretical quantiles figure first row second row samples size first column corresponds second third bandwidth intercept standard deviation equal obtained using naive bootstrap procedure estimated slope parameter standard deviation equal cox model estimated intercept slope respectively confidence interval given intercept slope variance based naive bootstrap procedure graph two estimators function given figure estimated coefficients curves quite close suggesting cox model might valid also confirmed figure shows estimation survival function uncured patients based estimation procedure procedure based cox model figure shows two estimators close values next analyse data provided medical birth registry norway see http data set contains information births norway since related total women interested time birth first second child mothers whose first child died within first year covariate interest age age mother birth first child age ranges years average years cure rate fraction women gave birth figure boxplots variance first row second row obtained bootstrap resamples samples size first column corresponds second third bandwidth empirical variance estimators also added dashed line figure shows estimator suggests cure fraction present first data set analyse data using approach proposed paper also using cox mixture cure model estimated intercept equals using model using cox model bootstrap confidence interval intercept estimated standard deviation equals estimated slope equals respectively using two models estimation procedure confidence interval given estimated standard deviation equal figure shows two estimators function quite different opposite slopes moreover survival function uncured patients given figure see estimator based cox model quite different suggesting cox model might valid data although formal test would need confirm however beyond scope paper also note estimator cure proportion increasing model decreasing cox model seems however natural believe probability second child cure proportion increasing age indication cox model valid data figure analysis breast cancer data estimator graph proposed estimator solid curve estimator based cox model dashed curve estimation using proposed estimator solid curve using estimator based cox model dashed curve idem appendix proofs proof proposition properties likelihood bernoulli random variable given log log integrate respect deduce log log exists last inequality becomes equality necessarily almost surely almost theorem deduce necessarily lemma let conditions hold true sup sup sup sup sup figure analysis second birth data estimator graph proposed estimator solid curve estimator based cox model dashed curve estimation using proposed estimator solid curve using estimator based cox model dashed curve idem defined similarly replaced fbc respectively moreover sup sup sup proof lemma let first investigate uniform convergence estimated cub let write mulative hazard measure well defined since probability tending integrals respect function bounded map variation uniform convergence assumption implies sup sup sup sup constant next duhamel identity see gill johansen fbc fbc uniform convergence fbc follows uniform convergence hence omit condition type arguments apply details next since conditions inf inf inf exists constant property inf inf inf fbc hence uniform convergence follows proof theorem let write moreover let log log log log let defined similarly let consider fbc respectively estimator equation replaced becomes arg max first step check sup follows directly lemma next given assumptions easy check assumption constant depending assumption positive values inf inf follows class hence sup finally proposition guarantees arg max gathering facts deduce proof theorem show asymptotic normality estimator verifying conditions theorem chen first consistency refer section whereas conditions chen satisfied construction thanks assumption respectively concerning first note expression inside expected value linear hence focus attention latter derivatives first using duhamel formula see gill johansen write similar way find finally note denominators bounded away zero thanks tedious rather elementary arguments follows formulae constant hence easily seen satisfies second property assumption chen hence holds true similarly decomposing using arguments first property assumption easily seen hold true next conditions satisfied thanks assumption follows calculations certain measurable functions remains verify condition note functions satisfying hence follows assumption theorem chen finishes proof proof theorem prove theorem check conditions theorem chen gives high level conditions naive bootstrap consistent difference setting setting proving bootstrap consistency whereas result holds true consequence high level conditions replace statements corresponding statements first follows assumption condition chen holds replaced neighborhood proof theorem follows holds true condition next conditions chen follow fact assume assumption continues hold remains verify condition true replace chen follows theorem chen whose conditions verified already theorem references akritas van keilegom nonparametric estimation residual distribution scand statist amico legrand van keilegom mixture cure model submitted boag maximum likelihood estimates proportion patients cured cancer therapy roy statist soc series chen linton van keilegom estimation semiparametric models criterion function smooth econometrica fang sun maximum likelihood estimation semiparametric mixture model scand statist farewell use mixture models analysis survival data survivors biometrics gill lectures survival analysis lectures probability theory ecole xxii lecture notes mathematics springer gill johansen survey view toward application survival analysis ann statist kuk chen mixture model combining logistic regression proportional hazards regression biometrika lin racine optimal bandwidth selection nonparametric conditional distribution quantile functions buss econ statist cao van keilegom nonparametric incidence estimation bootstrap bandwidth selection mixture cure models comput statist data anal lopez nonparametric estimation multivariate distribution function censored regression model applications commun stat theory meth maximum likelihood estimation proportional hazards cure model ann inst stat math maller zhou survival analysis long term survivors wiley new york meeker limited failure population life tests application integrated circuit reliability technometrics othus tiwari class semiparametric mixture cure survival models dependent censoring amer statist assoc peng taylor cure models klein van houwelingen ibrahim scheike editors handbook survival analysis handbooks modern statistical methods series chapter pages chapman hall boca raton usa schmidt witte predicting criminal recidivism using split population survival time models econometrics taylor estimation cox proportional hazards cure model biometrics taylor estimation failure time mixture models biometrics van der vaart asymptotic statistics cambridge university press wang klijn sieuwerts look yang talantov timmermans gelder jatkoe berns atkins foekens profiles predict distant metastasis primary breast cancer lancet peng nonparametric cure rate estimation covariates canad statist
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representation microseismic signals using synchrosqueezing transform roberto tary mirko van der baan university alberta edmonton canada rhherrer jan summary resonance frequencies provide useful information deformation occurring fracturing experiments management complementary microseismic event distribution accurate representation crucial importance prior interpreting cause resonance frequencies microseismic experiments popular methods fourier transform stft wavelet analysis limitations representing close frequencies dealing fast varying instantaneous frequencies often nature microseismic signals synchrosqueezing transform sst promising tool track resonant frequencies provide detailed representation apply synchrosqueezing transform microseismic signals also show potential general seismic signal processing applications introduction traditional time frequency representations fourier transform stft wavelet transform special representations like empirical mode decomposition emd limitations signal components well separated plane synchrosqueezing first introduced context speech signals daubechies maes shown alternative emd method daubechies improving spectral resolution hand improvements representation applied microseismic signals increase readability frequency spectrum auger flandrin identify independent components thakur accurate representation resonance frequencies could help gain better understanding fracturing process challenging determine exactly beneath earth surface hydraulic fracturing process paper demonstrate potentiality sst time frequency representation microseismic signals first describe underlying modulation model microseismic signals validate method synthetic examples finally apply sst real data sample high amplitude noise theory coherent signals microseismic recordings anthropogenic natural sources cases signal similar oscillatory characteristics could represented sum individual harmonic components cos instantaneous amplitude instantaneous frequency resonator represents additive noise including contribution environmental acquisition sources stands maximum number components one signal geoconvention integration thus microseismic signals show similarities representation tfr speech signals daubechies maes number harmonics components signal random appear different time slot different amplitude instantaneous frequency cwt sst cwt signal daubechies complex conjugate mother wavelet time shift applied mother wavelet also scaled cwt simply signal number wavelets scaled translated versions original mother wavelet coefficients representing concentrated picture used extract instantaneous frequencies daubechies wavelet coefficients often spreads around scale dimension leading blurred projection representation daubechies maes show smear neglected instantaneous frequency computed derivative wavelet transform point final step new representation map information timescale plane plane every point converted operation called synchrosqueezing daubechies since discrete values scaling step computed likewise mapping plane plane winst synchrosqueezing transform determined centers frequency range equation shows representation signal synchrosqueezed along frequency scale axis liang synchrosqueezing transform reallocates coefficients continuous wavelet transform get concentrated image timefrequency plane instantaneous frequencies extracted ultimate goal microseismic signal analysis identified frequencies used describe source mechanisms eventually gain better understanding reservoir deformation synthetic example following modulation model equation create noiseless synthetic signal sum following components brackets cos cos cos sin cos cos sin cos geoconvention integration synthetic signal figure two constant harmonics time component modulated sinusoid modulated signal central frequency appears time vanishes figure shows instantaneous frequencies component stft able identify four components low resolution see figure especially overlap hand sst figure able perfectly delineate individual component resolve instantaneous frequencies close theoretical value including amplitude modulated figure synthetic example synthetic signal four sinusoidal components instantaneous frequencies obtained derivative one independent components stft synthetic signal hanning window samples overlap used note smearing effect bandwidth sst shows sharper representation instantaneous frequencies application real microseismic signals section apply sst real dataset microseismic experiment recorded twelve geophones deployed vertical well stages sampling frequency receivers fluid injection located approximately depth sst challenged typical problems encountered microseismic experiments analysis time frequency resolutions noise stability capability identify localized frequency variations segment minutes data presenting clear resonance frequencies well sharp smooth changes frequency content chosen challenge performance sst representation stft used reference see figure upper plot stft calculated using window length overlap involving time resolution order four main resonance frequencies clearly visible approximately figure top two additional lines lower amplitude also visible lines approximately amplitude lines becomes weaker one almost vanishes line abruptly drops smooth changes barely discernible line fuzzy background due high amplitude noise present data also spectral leakage introduced fourier transform despite noise resonance frequencies well variations clearly visible sst representation figure bottom sst able map smooth sharp changes frequency lines compared stft sst brings resonance frequencies sharply improving significantly frequency resolution sst able geoconvention integration distinguish different lines constituting resonance frequency stft last least sst determines ifs times therefore time resolution method limited size window sst use bump wavelet ratio central frequency bandwidth discretization scales cwt sst produces sharper representation showing frequency components hidden stft representation figure real example upper plot shows stft lower plot sst sst able delineate spectral components missing stft representation conclusions introduced synchrosqueezing transform application microseismic signal analysis new transform shows promising results identification resonant components hence explanation microseismic phenomena enhancement spectral resolution allows separation close spectral bands fourier transform smears frequency components sst therefore attractive analysis microseismic signals acknowledgements authors thank sponsors blind identification seismic signals bliss project microseismic industry consortium financial support authors gratefully acknowledge anonymous company permission show use data references auger flandrin improving readability representations reassignment method ieee trans signal daubechies ten lectures wavelets society industrial applied mathematics presented regional conference series applied mathematics daubechies synchrosqueezed wavelet transforms empirical mode tool applied computational harmonic analysis daubechies maes nonlinear squeezing continuous wavelet transform based auditory nerve models wavelets medicine biology crc press liang generalized synchrosqueezing transform enhancing signal representation signal processing thakur brevdo synchrosqueezing algorithm spectral analysis robustness properties new paleoclimate applications mitre technical paper http flandrin daubechies one two frequencies synchrosqueezing answers advances adaptive data analysis aada geoconvention integration
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constructor rewrite systems ugo dal simone aug january abstract prove orthogonal constructor term rewrite systems weak reduction allowed scope reduction simulate linear overhead particular weak betareduction simulated orthogonal constructor term rewrite system number reduction steps conversely reduction term rewrite system simulated constant number steps relevant implicit computational complexity number beta steps normal form polynomially related actual cost performed turing machine normalization weak reduction orthogonal constructor term rewrite systems thus polynomially related turing machines taking notion cost natural parameters motivations implicit computational complexity young research area whose main aim description complexity phenomena based language restrictions external measure conditions explicit machine models borrows techniques results mathematical logic model theory recursion theory proof theory allowed incorporation aspects computational complexity areas formal methods software development programming language design developed area implicit computational complexity probably model theoretic one finite model theory successful way describe complexity classes design programming language tools type systems however syntactical techniques prove useful last years seen much work restricting recursive schemata developing general proof theoretical techniques enforce resource bounds programs important achievements characterizations several complexity classes means limitations recursive definitions recently using light fragments linear logic moreover rewriting techniques recursive path orderings interpretation method recently proved useful field borrowing terminology software design technology may dub area implicit computational complexity large aiming broad global view complexity classes may also implicit computational complexity small using logic study single models computation indeed many models computations come natural cost model definition cost intrinsically rooted model computation time polynomially related cost implementing model computation standard turing machine main example natural intrinsic parameter computation number parameter bears relation general actual cost performing computation since dipartimento scienze dell informazione bologna mura anteo zamboni bologna italy dallago dipartimento scienze dell informazione bologna mura anteo zamboni bologna italy martini may involve duplication arbitrarily big call implicit computational complexity small therefore gives complexity significance notions results computation models natural cost measures exist obvious particular looks simulations computational models present paper applies viewpoint relation orthogonal constructor term rewrite systems prove two machine models simulate linear overhead constructor term rewrite system could simulated well known view availability operators may used solve mutual recursion expressed rewrite rules section make explicit complexity content simulation showing rewriting steps simulated beta steps depends specific rewrite system size involved terms crucial result encoding constructor terms using scott schema numerals indeed parigot see also shows pure church numerals admit predecessor working constant number beta steps moreover splawski urzyczyn show unlikely encoding could work typed context system section studies converse simulation weak reduction means orthogonal constructor term rewrite systems give encoding constructor term rewrite system write map returning term given sense complete defunctionalization represented atomic constructor similar although technically use supercombinators show simulated step step rewriting theorem consequence taking number beta steps cost model weak equivalent linear function taking number rewritings orthogonal constructor term rewrite systems relevant implicit computational complexity small number beta steps normal form polynomially related actual cost performed turing machine normalization weak reduction established sands gustavsson moran fine analysis implementation based stack machine constructor term rewrite systems thus reasonable machines see invariance thesis taking notion cost natural intrinsic parameters byproduct section sketch different proof cited result instead using stack machine show could encode constructor term rewriting term graph rewriting term graph rewriting avoid explicit duplication substitution inherent rewriting thus also moreover exploit possible sharing subterms study complexity constructor graph rewriting relations constructor term rewriting found section show obtain results previous sections replaces underlying strategy paper extended version one title appeared proceedings icalp besides including full proofs extended section new material section preliminaries language study pure untyped endowed weak never reduce abstraction reduction definition following definitions standard full size duplicated term indeed arbitrary depend size original term reduction started situation much different weak reduction see terms defined follows ranges denumerable set denotes set assume existence fixed total order way sequence without repetitions variables set term said closed empty sequence values defined follows weak reduction denoted obtained closing reduction applicative context ranges terms ranges values length defined follows induction weak reduction enjoys many nice properties particular diamond property holds consequence number beta steps normal form invariant reduction order justifies way defined reduction slightly general plotkin one meaningful define time number beta steps normal form normal form exist cost model referred unitary cost model since beta weak reduction step counts global cost normalization moreover notice needed reduction closed terms closed reduced redex form closed value consequence arguments always closed open variables captured following lemma gives generalization combinator observe explicit limit reduction length spirit implicit computational complexity small lemma every natural number terms natural number sequence values proof terms looking simply following every natural number simply consider paper orthogonal constructor term rewrite systems crs see constructor term rewrite system pair symbols signature either constructors function symbols arity terms built constructors called constructor terms terms built constructors variables called patterns terms built constructor function symbols called closed terms terms built constructors functions symbols variables dubbed terms rules form function symbol consider orthogonal rewrite systems assume distinct two rules overlapping every variable appears lhs rule moreover assume reduction substitution triggering reduction must assign constructor terms variables restriction anyway natural constructor rewriting term crs denotes number symbol occurrences denotes number occurrences symbol constructor term rewriting definition crs constructor rewrite system defined set rules infinite signature particular signature includes binary function symbol app constructor symbols every every arity length every term associate term follows app observe closed rewrite rules rules following form app term canonical either app canonical notice signature contains infinite amount constructors example consider app moreover app expected finally term corresponds definition every term associate term follows happ hcx htn canonicity holds terms obtained images closed via moreover canonicity preserved reduction lemma every closed canonical moreover canonical canonical proof canonical induction structure hypothesis either abstraction application closed prove canonical whenever includes variables clearly canonical app canonical xim tim canonical implies rhs instance rule canonical consequence canonical whenever canonical concludes proof canonical terms normal form equivalent mapped normal form via true general take counterexample app corresponds via lemma canonical term normal form iff normal form proof canonical normal form contain function symbol app consequence abstraction always normal form conversely normal form form app otherwise closed application normal form since canonical contains terms normal form following substitution lemma useful later lemma every term every htn whenever includes variables proof induction hxi hti htn htn app happ happ htn htn htn happ htn htn hcy hcy hum htn hum htn htn hcy htn htn concludes proof lemma every proof induction happ hcy concludes proof previous two lemmas implies includes variables htn reduction simulated reduction provided starting term canonical lemma canonical proof consider instance rewriting rule turns let app clearly happ htn implies thesis conversely reduction simulated lemma canonical proof let redex fired rewriting must corresponding subterm app hcx observe definition hcx htn since canonical moreover since value implies app htn htn concludes proof previous lemmas altogether imply following theorem normalization mimicked reduction theorem term reducibility let closed term following two conditions equivalent normal form normal form proof suppose normal form applying lemma obtain term lemma canonical lemma normal form suppose normal form applying times lemma obtain lemma normal form lemma since canonical lemma another nice property crucial proving main result paper proposition every every every occurrence constructor subterm proof assume proceed induction example let consider notice clearly app moreover app app every constructor occurring term previous reduction sequence subterm remark order infinite crs since contains infinite amount constructor symbols moreover infinitely many rules consequence presented embedding weak infinite orthogonal crs consider following scenario suppose used write program suppose inputs form infinite set anyway represented finite set constructors scenario proposition allows conclude existence finite subsets every reduced via using constructors rules finite subsets consequence see schema one puts program correspondence finite crs finally observe assuming data representable finite number constructors reasonable scott scheme example allows represent term given free algebra finitary way natural number becomes becomes church scheme hand property constructor term rewriting section show rewriting step constructor rewrite system simulated fixed number weak let orthogonal constructor rewrite system finite signature let constructors let function symbols following constructions work independently first concentrate constructor terms encoding using scott schema constructor terms easily put correspondence way map defined induction follows hhci hhtn way constructors become functions hhci xar trivially hhci hhtn rewrites hhci steps represent error value use built way either form denoted metavariables like map defines encodings constructor terms terms containing function symbols goal defining another map returning given term way implies moreover rewrite whenever rewriting causes error whenever normal form containing function symbol first define corresponding constructor define mxi every every every variables induction mxi xar mxi xar lemma constant every every mxi hhtm hhtar hhci tar mxi hhtm xar whenever either hhtj least one among xar proof proceed induction mxi xar hhtar xar hhtar hhtar hhci tar use following abbreviations hhtm hhtm let distinguish two cases mxi hhtm xar xar xar let uar mxi hhtm xar hhcj uar xar hhuar xar hhtm zar hhtar xar hhtm hhcj uar xar mxi xar inductive hypothesis last term reduction sequence reduces correct normal form existence natural number prescribed properties proved observing none reductions length depends parameters hhtm xar concludes proof required lambda term simply interpreting function symbols difficult since embed reduction rules interpreting function symbol need preliminary result encode pattern matching lemma pattern matching let sequences patterns length term integer every sequence values hhpm tkmm hhtkmm whenever tji constructor terms moreover whenever unify sequences proof induction number constructors occurrences patterns inside always return simply sequence variables assuming term defined induction returns one first arguments otherwise returns argument applied first arguments must integers rar constructor patterns every every define sequences patterns values wpj follows rar defined sequence rar moreover simply indentity rar wpj undefined finally defined sequence xar wpj following hhcj xar number variables number variables consequence every find natural number sequence pairwise distinct natural numbers itj exactly sequences defined construction able formally define term zar wijt yitj ptj notice every moreover every length justifies application induction hypothesis concludes proof every function symbol let tni rules moreover suppose variables appearing patterns recall signature function symbols lambda term interepreting defined wij yar wini whenever moreover defined induction follows tar tar necessary ingredients extend mapping every term tar tar tar tar theorem natural number every function symbol every tar following three implications hold stands tar stands hhtar rewrites steps rewrites steps rewrites normal form rewrites diverges diverges proof easy combinatorial argument following definition clearly constant theorem depends independent particular term graph representation previous two sections proved main simulation result paper complete picture show section unitary cost model weak hence number rewriting constructor term rewriting system polynomially related actual cost implementing introducing term graph rewriting following adapting framework constructor rewriting contrarily section stay abstract attention restricted particular graph rewrite system needed implement reduction refer reader details efficient simulations term graph rewriting constructor term rewriting innermost outermost reduction strategies mentioned introduction see another proof means definition labelled graph given signature labelled graph consists directed acyclic graph together ordering outgoing edges node partial labelling nodes symbols node matches arity corresponding symbols labelling undefined formally labelled graph triple set vertices total ordering function partial labelling function length arity defined otherwise labelled graph closed iff total function consider signature arities respectively constructors examples labelled graphs signature following ones symbol denotes vertices underlying labelling function undefined consequence edge departs vertices role similar one variables terms one vertices labelled graph selected root obtain term graph definition term graphs term graph quadruple labelled graph root term graph following graphic representations term graphs root vertex drawn inside circle classes paths particularly relevant purposes definition paths path labelled graph said constructor path iff every symbol constructor pattern path iff every either constructor symbol undefined left path iff symbol function symbol pattern path definition homomorphisms homomorphism two labelled graphs signature function preserving term graph structure particular dom obvious generalization sequences vertices homomorphism two term graphs homomorphism two labelled graphs isomorphic iff bijective homomorphism case write similarly term graphs following consider term graphs modulo isomorphism iff observe two isomorphic term graphs graphical representation definition graph rewrite rules graph rewrite rule signature triple labelled graph vertices called left root right root respectively path starting left path following examples graph rewriting rules assuming function symbol constructors definition subgraphs given labelled graph vertex subgraph rooted denoted term graph subset whose elements vertices reachable appropriate restrictions definition redexes given labelled graph redex pair rewrite rule homomorphism vertex dom path starting constructor path last condition definition redex needed capture nature rewriting process given term graph redex result firing redex another term graph obtained successively applying following three steps build phase create isomorphic copy portion contained add obtaining underlying ordering labelling functions defined natural way redirection phase edges pointing replaced edges pointing copy root root newly created graph newly created copy graph obtained garbage collection phase vertices accessible root removed graph obtained write simply cause ambiguity case example consider term graph rewriting rule yttt homomorphism particular maps rightmost vertex applying build phase redirection phase get follows sss ysss finally applying garbage collection phase get result firing redex definition constructor graph rewrite system cgrs signature consists set graph rewrite rules term rewriting graph rewriting term signature turned graph obvious way tree vertices correspondence symbol occurrences conversely term graph turned term remember consider acyclic graphs similarly term rewrite rule signature translated graph rewrite rule follows take graph representing trees fact union two trees share nodes representing variable take root root example consider rewriting rule translation graph rewrite rule following arbitrary constructor rewriting system turned constructor graph rewriting system definition given constructor rewriting system corresponding constructor graph rewriting system defined class graph rewrite rules corresponding given term corresponding graph term graph corresponds term hgir let consider graph rewrite rules corresponding rewrite rules easy realize following invariant preserved performing rewriting whenever vertex reached two distinct paths starting root shared path starting constructor path term graph satisfying invariant said holds term graphs coming terms preserved graph rewriting lemma every closed term moreover closed proof fact every follows way map defined introduce sharing suppose corresponds term rewrite rule term graph obtained build phase obtained adding new nodes namely isomorphic copy portion contained notice stronger sense vertex reached newly created copy two distinct paths must constructor path consequence graph rewrite rule corresponding term rewrite rule shared vertices labelling function undefined redirection phase preserves one pointer redirected vertex labelled function symbol destination redirection vertex newly created copy edge incident clearly garbage collection phase preserve lemma closed term graph normal form iff hgir normal form proof clearly closed term graph normal form hgir term normal form redex translates redex hgir hand hgir normal form normal form redex hgir translates back redex reduction level graphs correctly simulates reduction level terms underlying graphs constructor shared lemma closed hgir hiir proof fact reduction step starting mimicked reduction steps hgir known literature redex term graph shared counterexample easily built consider term rewrite rule following term graph correspond term graph rewrites one step following one term rewrites two steps expected graph reduction even complete respect term reduction proviso term graphs must lemma hgir hiir theorem graph reducibility every constructor rewrite system every term following two conditions equivalent normal form normal form hgir proof suppose normal form applying lemma obtain term graph hgir lemma canonical lemma normal form suppose hgir normal form applying times lemma obtain hgir normal form lemma since constructor shared due lemma term rewrite systems graph reducible two conditions theorem equivalent see however othogonal constructor rewrite system graph reducible due strict constraints shape rewrite rules result considered analysis graph rewriting instrumental efficiently reduced graph rewriting corollary theorem theorem obtain possibility reducing term graphs purpose use cgrs corresponding corollary let closed term following two conditions equivalent normal form normal form however missing tales let analyze closely combinatorics graph rewriting consider closed term graph proposition lemma every constructor appearing label vertex subterm consequence difference big consequence exploit essential way possibility sharing constructors whenever computing graph takes polynomial time polynomially bounded hence theorem polynomial every normal form computed time time mentioned introduction achieved using explicit representations moreover reading back term graph take exponential time mentioned introduction complement theorem completeness statement universal computational model invariant cost model embedded polynomial overhead exploit analogous result proved theorem unitary cost model easily proved parsimonious difference cost model considered theorem let computed turing machine time suitable encoding pvq normalizes beta steps variations reduction purpose last section showing similar techniques applied evaluation previous sections endowed weak reduction technique however applied weak reduction sketch section endowed relation defined follows similarly case time stands number reduction steps normal form since relation deterministic functional time need another crs called similar designed simulate weak reduction signature includes binary function symbol app constructor symbols every every exactly moreover another binary constructor symbol capp every term associate terms follows capp app notice maps lambda terms constructor terms terms obtained via contain function symbols rewrite rules rules following form app capp app app app capp app app app ranges ranges abstractions applications rewrite rules said ordinary rules also need following administrative rule app capp app app ctrs slightly complicated additional overhead needed force reduction happen head position usual every term associate term happ hcapp hcx htn term canonical either app canonical lemma every closed canonical proof straightforward induction obvious variation equation holds htn mimics reduction much way mimics reduction however one reduction step corresponds steps although kept control lemma suppose canonical natural number canonical term whenever whenever proof term said iff app either element prove natural number canonical term whenever whenever proceed induction definition always form app distinguish three cases get want induction hypothesis form canonical form capp app capp app app apply induction hypothesis app since length strictly smaller proceed lemma since whenever rewrites one ordinary rules lemma canonical term normal form iff normal form proof first prove canonical normal form written proceed induction thesis holds app canonical normal form hence form induction hypothesis consequence normal form contraddiction prove statement lemma distinguishing two cases normal form abstraction hence normal form app normal form since canonical normal form consequence written concludes proof observe property holds canonical term reduce another one canonical even underlying normal form lemma canonical proof similar one lemma slight mismatch reduction reduction anyway harmless globally total number reduction step two times large total number reduction steps theorem term reducibility let closed term following two conditions equivalent normal form normal form moreover proof suppose normal form closed lemma canonical iterating lemma lemma obtain existence term normal form since normal form normal form iterating lemma obtain normal form graph rewrite system corresponding sense section exactly case computing normal form graph representation term takes time polynomial number reduction steps normal form theorem polynomial every normal form computed time time hand hope directly reuse results section proving existence embedding crss weak distinct normal forms two cases widely known however translation used simulate reduction reduction missing tale relative performances terms obtained via cps translation reduce normal forms number steps comparable number steps normal form original terms conjecture answer yes leave task proving future work conclusions shown cost models weak step unitary cost orthogonal constructor term rewriting rule application unitary cost linearly related since turn cost model polynomially related actual cost reducing turing machine two machine models considered reasonable machines endowed natural intrinsic cost models see also gurevich opus abstract state machine simulation level abstraction strong embeddings consider compositional equivalence model important technical result paper ongoing future work includes investigation much simulation could recovered either typed setting see difficulties case strong reduction reduce abstraction novel techniques developed since analysis performed present paper easily extended cases acknowledgments authors wish thank kazushige terui stimulating discussions topics paper references barendregt eekelen glauert kennaway plasmeijer sleep term graph rewriting bakker nijman treleaven editors volume parallel languages parle parallel architectures languages europe pages springerverlag erik barendsen term graph rewriting terese bezem klop vrijer editors term rewriting systems chapter pages cambridge univ press stephen bellantoni stephen cook new characterization polytime functions computational complexity ugo dal lago simone martini invariant cost model computability europe volume lncs pages springer ugo dal lago simone martini derivational complexity invariant cost model int work foundational practical aspects resource analysis fopara eindhoven ugo dal lago simone martini constructor rewrite systems icalp part volume lncs pages springer girard light linear logic inform yuri gurevich sequential asm thesis current trends theoretical computer science pages world scientific simon peyton jones implementation functional programming languages prentice hall daniel leivant ramified recurrence computational complexity word recurrence feasible mathematics pages marion moyen efficient first order functional program interpreter time bound certifications logic programming automated reasoning international conference proceedings volume lncs pages springer michel parigot representation data workshop computer science logic proceedings volume lncs pages springer michel parigot paul constant time reductions mathematical foundations computer science international symposium proceedings volume lncs pages springer gordon plotkin theoretical computer science detlef plump term rewriting systems application computer science pages sands gustavsson moran lambda calculi linear speedups essence computation complexity analysis transformation essays dedicated neil jones number lncs pages springer verlag zdzislaw splawski pawel urzyczyn type fixpoints iteration recursion international conference functional programming proceedings pages acm peter van emde boas machine models simulation handbook theoretical computer science volume algorithms complexity pages mit press christopher wadsworth unusual numeral systems seldin hindley editors curry essays combinatory logic lambda calculus formalism academic press
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mar matrix groups maximal cycle length power two guodong deng yun fan school mathematics statistics central china normal university wuhan china abstract every element matrix group similar permutation matrix called matrix group references showed matrix group contains maximal cycle maximal cycle generates normal subgroup length maximal cycle equals prime square prime power odd prime matrix group similar permutation matrix group paper prove matrix group contains maximal cycle maximal cycle generates normal subgroup length maximal cycle equals power similar permutation matrix group key words matrix group permutation matrix group maximal cycle introduction gld denote complex general linear group dimension degree consisting invertible complex matrices subgroup gld said matrix group dimension gld every conjugation permutation matrix say permutation matrix group gld permutation matrix called matrix group cigler showed matrix group permutation matrix group general matrix called maximal cycle email address yfan yun fan similar permutation matrix corresponding cycle permutation length cigler conjectured conjecture matrix group contains maximal cycle permutation matrix group let matrix group containing maximal cycle generates normal cyclic subgroup cigler proved dimension prime permutation matrix group proved permutation matrix group dimension square prime power odd prime paper prove theorem let matrix group dimension positive integer contains maximal cycle generates normal cyclic subgroup permutation matrix group necessary preparations proof theorem made section fundamentals group theory please refer section prove theorem case quotient group cyclic subgroup generated maximal cycle cyclic group see theorem section prove theorem case quotient group cyclic see theorem preparations sketch necessary fundamentals group theory number theory matrix theory formulate way suitable later quotations order element group denoted ordg ord short group known context hci denote cyclic group generated assume ord hci isomorphic additive group residue ring integer ring modulo mapping hci way properties cyclic group hci exactly corresponding properties additive group listed follows hci denotes multiplicative group consisting reduced residue classes case said generator cyclic group hci automorphism hci corresponding exactly one hci automorphism additive group hci normal hci homomorphic subgroup centg hci called centralizer hci centg hci hci hci said paper concerned cyclic group order correspondingly next lemma appeared lemma assume hci cyclic group order let cyclic subgroup order generated mapping homomorphism onto kernel hence generator cyclic group exactly generators mapped denote cyclotomic polynomial degree running primitive roots unity note lemma implies subscript runs primitive roots unity specific case since unique primitive root unity get known formula remark structure multiplicative group known see remark following hold iii study behaviors elements quite different note provided even subgroups following result lemma subgroup multiplicative group one following holds cyclic group order hence proof remark surjective homomorphism subgroup hence cyclic group order assume following take generator hki holds otherwise hence hki replace word holds integer denote valuation odd integer lemma let odd iii odd proof obvious since binomial coefficients get mod iii noting odd eqn obtain mod mod lemma let finite group generated hci normal cyclic subgroup order mod mod hence one following two holds case hci dihedral group order case hci generalized quaternion group order proof since hci quotient group isomorphic cyclic subgroup hri group hci let hcia centralizer hci hcia since iff iff mod suppose zero odd since odd thus hcia iff mod iff mod group order consisting two elements particular obtain conclusion next assume odd done suppose note integer positive integer direct computation following formula lemma iii odd integer mod taking eqn obtain replacing get holds finally assume odd mod iff mod iff mod eqn hcia hence hcia find integer lemma assume odd integer mod taking eqn obtain replacing get conclusion next lemma shows condition usually satisfied study lemma proposition matrix group maximal cycle hci also proof lemma lemma complex matrix chara denote characteristic polynomial chara det denotes identity matrix lemma lemma lemma following two equivalent similar permutation matrix diagonalizable chara case factor chara corresponds exactly one cycle decomposition permutation permutation matrix note cyclotomic polynomial degree stands divides get immediate consequence lemma follows corollary let matrix similar permutation matrix positive integers chara chara quotient cyclic section always assume maximal cycle let primitive root unity spectrum set eigenvalues eigenvalue denoted dimension complex space decomposed direct sum taking vector every index get basis play key convenience denote role study nal blocked matrix basis eqn transformation diagonal matrix please see lemma consider matrix respect basis eqn recall easy see permutes permutes lemma rjk length course subset ejk erjk basis subspace aejk erjk aerjk ejk invariant matrix respect basis ord course root unity replacing ejk aejk get ejk particular permutation matrix case permutes set cyclically similarly hci denote group generated lemma proposition proposition let notations choice basis eqn matrix permutation matrix matrix group permutation matrix group statement matrix permutation matrix equivalent saying set permutes elements set case call basis rest section contributes mainly finding basis partition disjoint union mod mod correspondingly basis eqn partitioned eqn decomposed direct sum two subspaces subspaces denote linear transformations restricted respectively correspondingly matrices respectively remark two elements always fixed two always subspaces note matrix group case since traces permutation matrices integers trace vector easy see hence obvious permutation matrix lemma assume maximal cycle choice basis eqn permutation matrix proof since length equals note ord conclusion follows eqn following lemma exhibits information case enough later quotations lemma assume maximal cycle matrix group dihedral group order one following two holds charac even odd case choice basis eqn permutation matrix charac even odd proof easy see partitioned follows correspondingly diagonally blocked follows lemma either prove contradiction suppose eqn suitable choice basis assume get chara lemma second factor right hand side hence get matrix follows characteristic polynomial follows charac lemma similar permutation matrix however assumption lemma says similar permutation matrix contradiction forces thus obtain easy check holds otherwise holds following consider case mod find basis eqn permutation matrix start investigation characteristic polynomial lemma let assume maximal cycle let subspaces eqn odd char odd odd char even proof assumption partitioned lengths number rjh eqn eqn assume eji eji aeji denote matrices restricted since lemma kji thus cik charai cik conclusion obviously true assume writing odd writing odd charai cik however primitive root unity lemma collection collection primitive roots unity appears multiplicity eqn obtain char odd lemma hence proved next assume odd lemma iii odd case noting done conclusion second step preparation induction let action multiplication acts map isomorphism sets lemma let notation lemma assume matrix group matrix group dimension maximal cycle proof since isomorphism mal cycle length note complete proof enough show similar permutation matrix let ord compute char char two cases case remark use lemma get char since similar permutation matrix lemma corollary char case odd lemma applied odd char even since similar permutation matrix lemma corollary odd even cases lemma conclude similar permutation matrix get enough information mod proceed two cases lemma let assume maximal cycle assume matrix group cases choice basis eqn permutation matrix proof lemma matrix group dimension maximal cycle obviously acts multiplication trivially diagonal matrix hence commutes maximal cycle dimension lemma ord thus suppose char however eqn char reached impossible equality sible case lemma already basis permutation matrix thus basis eqn obtained permutation matrix note mod eqn lemma eqn seen matrix group dimension maximal cycle odd charac even comparing lemma see suitable choice basis permutation matrix lemma assume thar maximal cycle choice basis eqn permutation matrix proof prove induction remark odd lemma basis eqn permutation matrix lemma matrix group dimension maximal cycle let since even see hence unique element order mod see remark lemma note induction basis eqn permutation matrix basis eqn permutation matrix theorem assume matrix group contains maximal cycle hci normal quotient group cyclic permutation matrix group proof assumption assume lemma hci lemma lemma assume remark prove theorem three cases case hence lemma replace assume lemma basis eqn permutation matrix lemma permutation matrix group case hence lemma assume lemma basis eqn permutation matrix lemma permutation matrix group case odd lemma assume lemma basis eqn permutation matrix lemma done quotient section consider matrix group maximal cycle hci normal quotient group cyclic prove fact permutation matrix group easy extend lemma follows lemma let maximal cycle subgroup hci normal choice basis eqn permutation matrices matrix group permutation matrix group proof let matrix transform basis set vandemond matrix formed basis respect basis obviously permutation matrix since permutes elements thus permutation matrix similarly permutation matrix respect basis every element permutation matrix lemma let written follows note allowed proof mod mod however since see mod mod set another starting get pair taking get another pair way till exhausted obtain listed pairwise lemma theorem assume matrix group contains maximal cycle hci normal quotient group cyclic permutation matrix group proof since hci see lemma isomorphic subgroup multiplicative group lemma lemma lemma assume first show suitable choice since quotient group commutative image coincides image hci note ord ord eqn lemma equality implies write noting obtain abc replacing still since see lemma condition lemma lemma exists basis eqn permutation matrix specifically see remark using notation lemma method eqn construct basis precisely choosing follows let lemma take vector ejt eqn ejt aejt ejt basis subspace permutation matrix since bejt see eqn take bejt basis subspace permutation matrix let basis eqn permutation matrix hand ejt ejt interchanges ejt since commutes get aejt bejt aejt interchanges aejt similarly interchanges ejt word permutation matrix course see remark complete proof enough show implies permutation matrix lemma permutation matrix group easy computation odd obviously hab matrix group since lemma lemma basis eqn permutation matrix particular see remark note implies done acknowledgments authors supported nsfc grant number references alperin bell groups representations gtm springerverlag new york bocong chen hai dinh yun fan san ling polyadic constacyclic codes ieee trans inform theory cigler groups matrices prescribed spectrum doctoral dissertation http cigler matrix groups linear algebra applications guodong deng yun fan matrix group maximal cycle prime square length linear algebra applications guodong deng yun fan matrix groups maximal cycle power odd prime length linear algebra applications robinson course groups theory gtm new york
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international journal software engineering applications ijsea july combining analytical hierarchy process genetic algorithm solve timetable problem ihab sbeity mohamed dbouk habib kobeissi department computer sciences faculty sciences section lebanese university lebanon abstract main problems school course timetabling time curriculum classrooms addition problems vary one institution another paper intended solve problem satisfying teachers preferred schedule way regards importance teacher supervising institute score according criteria genetic algorithm presented elegant method solving timetable problem ttp order produce solutions conflict paper consider analytic hierarchy process ahp efficiently obtain score teacher consequently produce ttp solution satisfies teachers preferences keywords timetable problem genetic algorithm analytic hierarchy process decision making introduction timetable problem ttp process scheduling sequence courses teachers students rooms satisfy set various constraints constraints differ based institutions involved schools universities institutions lebanese elementary schools important constraint consider teacher preferences may related criteria teacher age address etc example teacher seniority opportunity choose course schedule teacher priority select preferred schedule teacher criteria may reflection social life relationships urban lebanese institutions note criteria may conflicting ranking teachers accordingly becomes obvious example case comparing senior teacher younger teacher analytic hierarchy process ahp developed saaty effective tool dealing complex decision making processes therefore building ranking relationship teachers based series pairwise comparisons addition ahp incorporates useful technique checking consistency decision evaluation hence reducing bias decision making process hand manual solution ttp still used nowadays reasons related lack budget needed buy dedicated ttp solver asc prime doi international journal software engineering applications ijsea july also inconsistency tools lebanese teachers criteria majority existing tools person responsible creating schedule must manually specify teachers priorities tool generate timetable schedule however teachers priorities always evident significantly measured addition ttp problem manual solution time effortconsuming problem would presented great number variables make intractable motivation approach means genetic algorithm paper present new approach combines ahp create time table schedule matches teachers preferences along institution using ahp develop teachers ranking providing score teacher satisfaction function incorporated produce schedule matches best ahp ranking satisfies teachers preferences informal description approach presented basing simple example formal description implementation method would future challenge rest paper structured follows section presents background works related utilization solve ttp section describes steps followed produce time table schedule taking consideration teachers preferences also present simple case study application procedure section ahp applied set teachers score calculation procedure presented section describes result combination ahp generate time table schedule section concludes paper describes ongoing works background genetic algorithm genetic algorithm type searching algorithm searches solution space optimal solution problem key characteristic genetic algorithm searching done algorithm creates population possible solutions problem lets evolve multiple generations find better better solutions population collection candidate solutions considering course algorithm generations algorithm new members born population others die population single solution population referred individual fitness function individual presents measure good solution represented individual better solution higher fitness obviously dependent problem solved selection process analogous survival fittest natural world individuals selected breeding based upon fitness values fitter individual likely individual able reproduce occurs mingling two solutions together produce two new individuals generation small chance individual mutate change individual small way information refer related works various types timetable problems appear regards educational institutions depend elements scheduled exams regular courses excellent survey exam timetabling techniques presented examination timetabling similar international journal software engineering applications ijsea july institutions consists scheduling exams set courses limited time period avoiding overlapping exams student well seeking largest spread examination period however course timetable differs regarding institution universities schools course timetable scheduling problem basically deals effective distribution five kinds entities resources rooms courses teachers students time days hours university timetables problem presents large number constraints satisfied constraints typically cover time conflict room conflict course conflict criteria related teachers students preferences commonly divided two types hard constraints soft constraints hard constraints violated one course allowed timeslot soft constraints may violated purpose lies minimizing violation example soft constraints university timetable student attend one course per day wave works proposed solutions university timetable problem underline works schools ttp may defined subset universities ttp university ttp number set constraints larger several works addressed school ttp presented different approaches solve problem highlight works works proposes different utilization definition parameters chromosomes description mutation crossover definition etc works apply parallel implementation solve school ttp however addressing teachers preferences way presented paper never previously carried would contribution process problem description common classic definition fitness function presented previous works takes consideration time conflict hence approach assumes time conflict already solved using one fitness functions previously mentioned concern consider teachers preferences feasible solution one satisfies preferences process described figure first ahp used calculate score teacher arising set information given according institution valuable criteria scores considered one input fitness function practically composed two functions conflict function satisfaction function feasible solution validated functions however omit process used conflict function may one previous mentioned works satisfaction function teachers scores parameters checks produced solutions satisfy teachers preferences given input figure process international journal software engineering applications ijsea july rest paper process applied simple case study described description school ttp paper present informal description approach purpose simplify problem description presentation solution later consider following simple ttp model school open days week daily sessions scheduled time slots two separated break session school provides classes thus class sessions per week total sessions achieved weekly school teachers school teachers distribution set sessions teachers given following table table table distribution sessions set teachers teacher total total note study take consideration course name math sciences literature etc therefore need specify course given session course assigned one teacher accordingly teacher schedule implemented without taking care course name recalling purpose simplify presentation approach model simple description reality standard lebanese school actually provides nine levels classes may one sections depending number students tolerable level case two sections level may consider section new separate class school usually open five days week offers daily six sections per class thus least total sessions scheduled nine classes set teachers back simple model table presents empty time slot table assigned teachers table empty time table international journal software engineering applications ijsea july previous related works two kinds constraints respected time table schedule hard soft restrict study necessary constraints constraints class two consecutive sessions teacher acceptable considered integration approach seems promising hard constraints time conflict teacher schedule teacher two sessions parallel time slot teacher assigned specified number session class soft constraint one soft constraint underlined related teacher preferences teacher would specify proffered schedule goal satisfy teachers preferences teachers preferences satisfied perfect time table schedule obtained table preferred schedule teachers mentioned described hard constraints already addressed previous listed works paper suppose able produce time table without time conflict teacher assigned exact number sessions required respecting soft constraint key issue intend solve satisfying teachers preferences order assume teacher provide preferred schedule institution combined algorithm generate time table satisfies teachers preferred schedule according satisfaction threshold fixed institution satisfaction function applied possible solution generated hence solution feasible greater simple case table presents preferred schedule six teachers value means specified session preferred teacher example problem schedule however prefers give sessions break session break session next section show generate score teacher using ahp scores crucial parameters satisfaction function international journal software engineering applications ijsea july analytical hierarchy process ahp simple tool deals complex problems developed thomas saaty widely known method ranking decision alternatives selecting best one decision maker multiple objectives criteria selection process based calculation scores alternatives key point ahp method pairwise comparison used calculate relative weights criteria consequently develop overall ranking alternatives order help pairwise comparison saaty created scale importance two elements interested five scales suggested numbers express degrees preference two elements shown table table standard scale used ahp preference level equally preferred moderately preferred strongly preferred strongly preferred extremely preferred numerical value approach ahp used order generate score teacher goal present steps followed ahp method applying simplified ttp example case study interested ranking list teachers alternatives according list criteria supposed defined school differ institution another simplify presentation approach consider four criteria teacher age teacher timer teacher gender teacher teaching load criteria related teachers health conditions marital status distance living address school may also significant teaching load teacher also essential data needed later approach even considered teacher criteria table presents list information teachers case study according criteria set criteria school specify importance criterion compared another criterion order importance translated value using ahp scale table table information teachers teacher age gender contract full full part part full full load international journal software engineering applications ijsea july consider contract type teacher load important factors give teacher superiority choose preferred schedule teacher primacy also teacher greater load prioritized teacher smaller load teacher age comes next importance criteria ages divided intervals intervals older ages become important preference level older teachers given priority choose course schedule younger teachers teachers age interval preference value teacher gender least importance preference order assume female teacher priority preference level according ahp scale table relative importance criteria described pairwise comparison rating criterion given table table pairwise comparison four criteria age teacher teacher load gender teacher contract type teacher per criterion age consider ascending exploration intervals correspond exploration preference scale thus age interval strongly preferred age moderately preferred age example pairwise comparison gives value meaning age strongly preferred age pairwise comparison according criterion gender respects fact female moderately preferred male teacher pairwise comparison criterion achieved according fact strongly preferred finally applied pairwise comparison load criteria basing division load value intervals strategy used criterion age applied international journal software engineering applications ijsea july note criteria teacher compared equally preferred compared criterion preference value preference value comparing second step ahp process generation preference vectors criterion table present calculate preference vector criterion gender table calculation preference vector criterion gender gender gender teacher teacher total total gender teacher average total first sum column calculated element column divided corresponding sum finally preference vector calculated value corresponds row average therefore preference vectors criteria shown table table preference vectors four criteria teacher age gender load contract international journal software engineering applications ijsea july third step ahp ranking criteria determine relative importance weights pairwise comparison used purpose agreeing relative importance previously described table shows criteria weights calculated similar way preference vectors table criteria weights criteria age gender load contract weight age gender load contract total score teacher may calculated multiplication weight table matrix table obtained result follows score score score score score score set score used next section incorporate overall satisfaction function order produce requested time table satisfying teachers preferences finally mention ahp also incorporates useful technique checking consistency decision evaluation consistency test may applied study verify considered pairwise comparisons reliable information consistency test refer combining ahp presented genetic algorithm proposes fitness function used test feasibility produced solution classic fitness function applied solve ttp takes consideration time conflict define function fconflict practically fconflict may return number conflicts generated solution solution feasible value fconflict task implemented several previous works approach another function fsatisfaction satisfaction integrated fitness function fsatisfaction score teacher generated ahp number sessions teacher match preferred schedule see table observe maximum value fsatisfaction called maxfsatisfaction load hence obtain perfect satisfaction international journal software engineering applications ijsea july result say solution feasible two following conditions respected fconflict fsatosfaction satisfaction threshold set designer note order obtain perfect satisfaction need set case study maxfsatisfaction table presents randomly generated solution fsatisfaction feasible assume table solution finally note feasible solution perfect satisfaction may generated case study shown table table feasible solution perfect satisfaction conclusion paper presents new approach deals teachers preferences constructing time table school schedule approach consists integration satisfaction function genetic algorithm parameters satisfaction function teachers loads set scores calculated using analytical hierarchy process key point ahp calculating scores pairwise comparison set teachers criteria new approach consequently combination ahp gives rise new methodology solve time table problem call paper present informal description approach formal description implementation procedure perspective addition need verify contribution applying implemented different case studies lebanese schools moreover consideration sophisticated constraints related international journal software engineering applications ijsea july courses order integration another important perception declaration set rules manual interception building time table also suitable references saaty analytic hierarchy process mockus jonas lina algorithms optimization high school timetables informatica asc timetable tool official website http prime timetable tool official website http holland john genetic algorithms optimal allocation trials siam journal computing thede introduction genetic algorithms comput sci colleges vol goldberg david edward genetic algorithms search optimization machine learning vol reading menlo park burke mccollum ltg merlot lee survey search methodologies automated system development examination timetabling journal scheduling gyori petres genetic algorithms timetabling new approach budapest university technology economics hungary ghaemi sehraneh mohammad taghi vakili ali aghagolzadeh using genetic algorithm optimizer tool solve university timetable scheduling problem signal processing applications isspa international symposium ieee abdullah salwani hamza turabieh generating university course timetable using genetic algorithms local search convergence hybrid information technology third international conference vol ieee chaudhuri arindam kajal fuzzy genetic heuristic university course timetable problem international journal advance soft computing application colorni alberto marco dorigo vittorio maniezzo genetic algorithm solve timetable problem politecnico milano milan italy dhanabalan nagarajan sakthivel genetic algorithm based timetabling approach international journal research reviews computer science abramson david abela parallel genetic algorithm solving school timetabling problem division information technology csiro carrasco marco margarida pato multiobjective genetic algorithm timetabling problem practice theory automated timetabling iii springer berlin heidelberg authors ihab sbeity main author associative professor computer science faculty sciences lebanese university received applied mathematics lebanese university master computer science systems communications joseph fourier university france phd inpg institut national polytechnique grenoble france phd works related performance evaluation system design actually research interests include software performance engineering uml modelling decision information system international journal software engineering applications ijsea july mohamed dbouk full time professor lebanese university coordinates research degree idss information decision support systems received phd france research interests include software engineering information systems related issues geographic information systems habib kobeissi associate professor data structure faculty sciences lebanese university since holds bachelor science computer science grenoble university master science data structure university paris data structure university paris worked computer scientist several french companies
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ijcsis international journal computer science information security vol mining spatial gene expression data using negative association rules department computer science engineering smit majitar india department drug technology higher institute medical technology derna libya rather based items within database associations items commonly expressed form association rules general association rule represents relationship two sets items database written form item sets side lhs rule called antecedent rhs called consequent negative association rules complementary sorts association rules forms rule form equivalent positive association rule form years data mining attracted attention research community researchers attempt develop faster scalable algorithms navigate ever increasing volumes spatial gene expression data search meaningful patterns association rules data mining technique tries identify intrinsic patterns spatial gene expression data widely used different applications lot algorithms introduced discover rules however algorithms used find positive association rules contrast positive rules negative rules encapsulate relationship occurrences one set items absence set items paper algorithm mining negative association rules spatial gene expression data introduced algorithm intends discover negative association rules complementary association rules often generated priori like algorithm study shows negative association rules discovered efficiently spatial gene expression data paper attempt made study novel algorithm discovering positive association rules generating meaningful negative association rules effective manner spatial gene expression data spatial gene expression data association rule negative association rule means iii introduction heading main contribution great explosion genomic data recent years due advances various biotechnologies spatial gene expression database large genomic data sets often contain much information researchers generated data might anticipated enormous data volume enables new types analyses also makes difficult answer research questions using traditional methods analysis massive genomic data two important goals notations means union means negation materials methods spatial gene expression data edinburgh mouse atlas gene expression database emage developed part mouse gene expression information resource mgeir collaboration jackson laboratory usa emage http freely available curated database gene expression patterns generated situ techniques developing mouse embryo spatial gene expression data presented similarity matrix element matrix measure similarity corresponding probe pattern region similarity calculated fraction overlap two total areas images measurement intuitive commonly referred jaccard index pattern compared jaccard value two input spatial regions identical compared another pattern jaccard index less one jaccard index two patterns intersect jaccard index value close two patterns similar determine expression particular gene might affect expression genes determine genes expressed expressed result certain cellular conditions genes expressed diseased cells expressed healthy cells popular pattern discovery method data mining association rule mining association rule mining introduced aims extract interesting correlations frequent patterns associations casual structures among sets items transaction databases data repositories relationships based inherent properties data however biologists interested gene expression changes different probe patterns thus study carried part research promotion scheme rps project aicte govt india corresponding author http issn ijcsis international journal computer science information security vol frequent itemsets two subsets also frequent support already calculated two subsets may generate rule confidence rule greater equal specified minimum confidence similarity values discretized similarity measure greater predetermined thresholds converted boolean matrix data preprocessing preprocessing often required applying data mining algorithms improve performance results preprocessing procedures used scale data value either values contained spatial gene expression matrix transformed boolean values socalled discretization phase context quantitative value given rise effect discretization procedure max minus method algorithm details let set items item corresponds value attribute member attribute domain binary attribute dom max minus procedure consists identifying highest expression value data matrix defining value expression gene given data expression value integer value otherwise expression gene assigned value figure similarity matrix items genes data set transaction consists genes expression pattern intersecting probe pattern sets transactions constructed taking probe pattern every gene associated gene expression pattern satisfies minimum similarity similarity form itemsets input discretization figure results max minus discretization method association rule mining algorithms adopt iterative method discover frequent itemsets process discovering frequent itemsets need multiple passes data algorithm starts frequent maximum frequent itemsets discovered algorithms consist two major procedures join procedure prune procedure join procedure combines two frequent kitemsets generate itemset new preliminary candidate following join procedure prune procedure used remove preliminary candidate set itemsets whose frequent itemsets transaction database database containing transactions form dom let transaction database number transactions minsup minimum support defined minsup proposition according boolean vector operation sum row vector smaller necessary involve calculation supports proposition according suppose itemsets kitemsets presents number items frequent set item smaller itemset frequent itemsets proposition presents number frequent set smaller maximum length frequent itemsets positive association rule represents relationship two sets items form positive association rule represents relationship two sets items form rule support data sets transactions contain itemset contain item set support negative association rule supp frequency occurrence transactions item set absence item set let set transactions contain items rule holds given data set confidence transactions contain item set confidence negative association rule conf calculated supp given supp conf support confidence negative rule computed follows supp supp supp conf conf given supp conf support confidence negative rule computed follows supp supp supp every frequent itemset two subsets constructed way one subset contains exactly one item remaining items subset downward closure properties http issn ijcsis international journal computer science information security vol part algorithm given figure capable discovering possible set frequent itemsets subject user specified minimum confidence conf supp introduced algorithm finding rhs negative lhs negative rules terms spatial gene expression data form similarity matrix consists four phases follows part algorithm given figure capable finding positive negative association rules frequent itemsets subject user specified minimum confidence quickly function genrule freqsetk generates positive subject user specified minimum confidence function gennegcand generates negative itemsets given positive association rule using support value calculated using formula given equations generated negative item sets negative association rules generated subject confidence calculated using formula given equations transforming similarity matrix boolean matrix generating set frequent itemsets using fast mining algorithm spatial gene expression data generating positive rules using apriori algorithm generating negative rules based existing positive rules detailed description introduced algorithm described follows part algorithm generating frequent itemsets positive rules results discussion introduced algorithm implemented java tested linux platform comprehensive experiment spatial gene expression data conducted study impact normalization input spatial gene expression data similarity matrix minimum support output set frequent itemsets normalize data matrix transformed boolean matrix frequent generation column sum else delete proposition row sum delete proposition join procedure produce combination columns combination sum prune procedure item delete column according item row sum delete return sample records spatial gene expression data emage listed table different thresholds tested positive negative rules listed table generated support confidence constraints respectively note rules right column negative rules discovered respect positive rules left column table shows number positive rules negative rules minimum support minimum confidence thus algorithm successfully generate negative rules number negative rules discovered reasonable number negative rules tends related number positive rules given table however inversely proportional minimum support threshold reason less less set survives increases support threshold reduces number candidate negative rules significantly table spatial gene expression data eamge database uniqid emage gene name emage emage emage emage table positive negative rules generated part algorithm generating positive negative association rules input set frequent minimum support minimum confidence output set positive negative association rules postiverule genrule freqsetk rule postiverule generate negative rules rules postiverule negativerulesets gennegcand rules negaitverulesets rule rule neg neg minsup neg minconf endif endfor fig mining negative association rules positive rules confidence negative rules confidence given number positive rules complexity algorithm algorithm complexity depend number transactions since assumed supports item sets counted stored use well mining applications however considering discovering positive rules necessary http issn ijcsis international journal computer science information security vol generating negative rules algorithm must browse combinations items complexity discovering positive rules depends number transactions also sizes attribute domains well number attributes overall complexity proportional discovering positive rules association rule mining ensures genomic data mining continue necessary highly productive field foreseeable future references baldock bard burger burton christiansen feng hill houghton kaufman rao emage framework understanding spatially organized neuroinformatics vol tan micahel steinbach vipin kumare intoduction data mining pearson education second edition agrawal srikant algorithms mining association rules large proceedings international conference large databases santiago chile agrawal imielinski swami association rules sets items large proceedings acm sicmod conference management washington zhang optimization algorithm base apriori association computer engineering vol van hemert baldock spatial gene expression data association hochreiter wagner editors proceedings international conference bioinformatics research development lecture notes bioinformatics springerverlag becquet mining data analysis case study human sage genome biology vol november spatial gene expression data using association ijcss vol venkataraman stevenson yang richardson burton perry smith baldock davidson christiansen mouse atlas gene expression update nucleic acids research jan van hemert baldock matching spatial regions combinations interacting gene expression patterns elloumi linial murphy schneider toma editors proceedings international conference bioinformatics research development volume communications computer information science pages springer verlag performance also affected choice minimum support lower minimum support produces numerous item sets confidence constraint positive rules generated adds computation expense trend number negative number positive rules different minimum support shown figure figure positive negative rules discovered different supports confidences conclusion paper novel method mining positive negative association rules spatial gene expression data introduced generate frequently occur genes quickly introduced algorithm produce candidate itemsets spends less time calculating itemsets boolean matrix pruned scans database needs less memory space compared apriori algorithm introduced algorithm good enough generating positive negative association rules spatial gene expression data fast memory efficient finally large rapidly increasing compendium data demands data mining approaches particularly http issn
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signal processing wireline communications jun zafaruddin member ieee itsik bergel senior member ieee amir leshem senior member ieee faculty engineering university ramat gan israel email smzafar leshema processing played important role improving quality communications copper cables earlier dsl technologies even powerful signal processing techniques required enable gigabit per second data rate upcoming standard new standard different predecessors many respects particular use significantly higher bandwidth high bandwidth crosstalk different lines binder reach unprecedented levels beyond capabilities efficient techniques interference mitigation article survey state art research challenges design signal processing algorithms system focus novel research approaches design considerations efficient interference mitigation systems also detail relevant vdsl techniques points strengths limitations system ntroduction digital subscriber lines dsl evolved viable technology last mile access telecommunication networks technology leverages existing infrastructure telephone lines provide affordable broadband services deployment optical networks unfeasible costly since inception dsl considered interim technology fill gap advent optical access networks however far away horizon time optical access network arrived yet dsl thus remains widely used broadband access technology play key role convergence mobile fixed technologies next generation networks upcoming fast access subscriber terminals standard promises achieve data rates exploiting higher bandwidth telephone lines previous standards expected deliver gigabit speeds short loop lengths anticipated cioffi decade ago new standard different predecessors many respects fiber deployed network rewiring houses fiber extremely expensive fiber distribution point fttdp technology taking fiber distribution point close customer premise equipment cpe new standard changed fundamental design choices used vdsl earlier standards notably technologies duplexing tdd rpf used discontinuous operation incorporated energy efficient transmissions tdd scheme avoids crosstalk next facilitates asymmetry downstream upstream data rates efficiently also simplifies channel estimation downstream transmissions exploiting channel reciprocity channel state information csi rpf technology simplifies powering using power customer cpe eliminates need power infrastructure reduces deployment costs discontinuous operation optimizes energy consumption incorporating low power modes well switching circuitry corresponding users offline mode system requires efficient signal processing techniques harness benefits features also uses considerably wider bandwidth vdsl system using spectrum mhz first generation system uses mhz whereas next generation goes mhz higher bandwidth systems suffer significantly stronger crosstalk due electromagnetic coupling different lines increasing frequency crosstalk coupling lines attains strength direct path destroys diagonal dominance channel strong crosstalk poses challenges resolved existing vdsl technology hence system requires newer powerful techniques interference mitigation also known crosstalk cancellation signal processing techniques played important role improving quality communications copper cables hold key future services article provides overview current state art research challenges design signal processing algorithms system focus crosstalk cancellation schemes signal processing referred vectoring ginis cioffi enabled vdsl systems exceed mbps signal processing takes place central point concentrates copper wire pairs many users processing signals transmitted central point end users signal received users thus vectored system resembles mimo standard mimo however vectored dsl system different wireless mimo important respects dsl channel long channel coherence thus csi estimated fairly efficiently processing since users connected fixed modem channel tracking also difficult moreover telephone channel offers fairly good channel signal noise ratio snr high ireline dmt echnology like communication system performance dsl systems limited several types impairments following discuss main ones describe traditional novel techniques implemented dsl systems address also discuss key distinguished features technology show different predecessor technologies thermal noise communication systems inherently limited thermal noise caused random movement electrons system additive noise generally modeled random gaussian signal independent transmit signal effect thermal noise completely avoided sets upper limit communication performance known channel capacity attaining channel capacity requires implementation powerful error frequency selective channel frequency flat channel gain low frequencies fact dsl systems baseband eliminates detrimental effect phase noise therefore allows high spectral efficiency reach qam standard uses much higher frequencies predecessor vdsl standard longer rely key features vdsl one features well conditioning channel matrix vdsl frequencies dsl channel matrix diagonal dominance cwdd upstream diagonal dominant rwdd downstream characteristic convenient fext cancellation techniques interference mitigation vdsl rely diagonal dominance channel matrix diagonal dominance enables efficient implementation well linear crosstalk cancelers linear zero forcing precoder downstream linear canceler upstream near optimal vdsl channel schemes simple necessarily require transmit power optimization even matrix inversion required linear crosstalk cancellation avoided power series expansion mimo channel simple least mean square lms based adaptive algorithms efficient converge quite rapidly diagonally dominant vdsl channels high frequencies crosstalk significant channel matrices longer diagonal dominant thus many vdsl algorithms either fail converge slowly moreover adaptive schemes vdsl system either suitable longer applicable tdd based system paper broad scope begin overview single user signal processing explain enables dsl modem overcome channel impairment provide overview channel explain various alternative techniques crosstalk cancellation conclude design considerations provide insight techniques used zoomed frequency mhz fig frequency selectivity dsl channel illustrated using channel gain cable simulated profile figure shows use tones khz result almost frequency flat channel tone correction codes legacy dsl systems employ relatively simple reed somolon codes outer code trellis coded modulation tcm inner code interleaver combined coding scheme also chosen mhz standard ldpc codes also proposed however design implementation ldpc code flexible coding modulation operate data rates open research challenge however scenarios thermal noise main limiting factor impairments must considered interference dmt modulation telephone channel frequency selective exhibits frequency dependent attenuation delay causes severe interference isi communication symbols prolonged overlap overcome problem recent xdsl standards use discrete dmt technique divides transmission frequency band smaller also known tones frequency bins result dmt effectively transforms broadband frequency selective channel many narrow band channels shown fig wider tone width exactly times wider cover higher bandwidth without increasing number tones adsl system tones mhz vdsl mhz contains tones mhz tones mhz profile increases number tones dmt symbol also contains cyclic prefix allows tone separation symbol duration without interference tones long longer channel memory see example typical length samples time domain samples samples tones input bits channel encoder dmt encoder ifft dac mhz khz output bits dmt decoder channel remove mhz adc mhz mhz fig discrete dmt modulation system parameters system processing external interference impulse noise rfi etc upstream mac external interference impulse noise rfi etc port cpe direct path cpe cpe next port fext fext next port direct path twisted pairs downstream cable binder tdd khz time fdd vdsl fdd vdsl time fig duplexing methods tdd system profile fdd profile vdsl separate upstream downstream transmissions mitigate next dsl system typical tdd frame duration comprising symbols number symbols downstream ranges symbols upstream single guard symbol binder port cpe fiber line mhz mhz fft mhz channel decoder note scale figures freqency add joint processing binder section fig fext next dsl binder cross section cable binder also shown demonstrate twisted pairs enclosed binder upstream dsl binder acts multiple acces channel mac whereas broadcast channel downstream note cpes generally situated different lengths higher sampling frequency vdsl system operates msps million samples per second sampling frequency mhz profile msps convert data symbols real signal point ifft used resulting bandwidth mhz considering resulting symbol rate khz exactly times faster mhz system symbol rate much higher sampling frequency rate msps typical dmt block diagram parameters represented fig another important advantage dmt ability use different modulation tone thus tones low snr use small constellations qpsk bits per tone tones high snrs use richer modulations bits per tone points dmt technique allows transmission data symbols without isi without interference different tones thus enables independent processing tone without loss generality following focus single tone address factor limit system performance crosstalk next duplexing methods telephone line composed two copper lines twisted around twisting reduces electromagnetic leakage lines however sufficient dsl systems cope electromagnetic coupling signals increase continuously frequency together large number closely packed lines typical binder lead large electromagnetic couplings cause significant crosstalk depending upon position disturbers respect victim receiver crosstalk classified crosstalk fext crosstalk next shown fig crosstalk next refers coupled signals originate end affected receiver hence next interference upstream signals downstream signals different pairs next occurs short loops effect receivers significant next mitigated using echo canceler considered impractical dsl setup thus dsl systems separate upstream downstream avoid next echo signals older standards adsl vdsl separate upstream downstream frequency domain known frequency domain duplex fdd latest standard extends copper bandwidth hundreds mhz employs duplexing tdd upstream downstream transmitted different times tdd scheme access full operating spectrum toggling transmission directions time interval occurs rapidly visible user enables dynamic allocation resources efficiently support asymmetric data rates facilitates discontinuous operation allows efficient trade throughput power consumption tdd scheme also ensures channel reciprocity better service provisioning facilitates channel estimation downstream also enables simplified analog afe architecture increases efficiency transmissions however successful implementation tdd needs precise timing synchronization system transmitter receiver avoid interference two directions requires nearby modems synchronized feasible situation generally customer premises tdd fdd duplexing systems illustrated fig samples samples dmt encoder user ifft user dac adc user channel post canc tone vectored users tone dmt encoder user post canc tone ifft user dac adc per tone feq vectored users tone cpe cpe fft user fft user user transmission cpe per tone feq vectored receive processing fig schematic diagram vectored receive processing upstream signal coordination among cpes joint transmit processing received signals cpes tone collected single vector canceler applied samples dmt encoder user ifft adc dac user vectored users tone vectored users tone fft user channel cpe cpe ifft user adc dac fft user per tone feq vectored transmit processing precoder tone precoder tone dmt encoder user per tone feq vectored reception shown fig tone values users collected form vector used coordinated signal processing components result processing fed conventional single user modem demodulation detection fig tone values users collected form vector used coordinated transmit signal processing components following describe vector processing focus single frequency tone time hence without ambiguity drop tone index quantities section note processing frequency bins identical performance different frequency bins significantly different resulting vector channel model upstream processing cpe fig schematic diagram vectored transmit processing downstream transmit signals user tone collected single vector precoder applied transmission signal coordination among cpes joint receiver processing crosstalk vectored processing term fext refers coupled signals originate end opposite affected receiver fext thus interference upstream signals different pairs downstream signals different pairs average fext power line ith line described dij ext constant whose value depends physical properties copper cable denotes attenuation disturber central frequency considered tone dij coupling length victim distributer fext creates linear dependence different lines binder hence calls joint vector processing upstream since modems possible construct vector contains received symbols lines given tone process together reduce effect fext hand downstream transmitters colocated possible signals reduce crosstalk coordinated processing signals lines referred vectoring leads enormous rate increase dsl system schematic diagram vector transmitted symbols matrix complex channel gains considered frequency tone represents additional noise interference vector note diagonal elements correspond direct paths cpes whereas elements represent fext downstream joint processing performed transmission thus channel structure remains vector symbols transmission lines joint processing vector samples received different cpes note tdd channel reciprocity dictates channel matrix downstream matrix transpose channel matrix upstream cases differences amplitude phase transmission reception circuits filters end fext currently main limiting factor dsl systems plays even crucial role upcoming technology various methods cope fext described following sections discussion channel model next section iii dsl hannel odels signal processing methods communication system require accurate characterization underlying transmission medium dsl technology modeling twisted pair channels higher frequencies active field research two decades dsl standards evolved resulting models based extensive measurement campaigns carried different laboratories telecommunication companies derive parametric cable models models direct crosstalk channels loss signal strength transmitted cable depends physics cable implicitly captured propagation constant loop length cable insertion loss modeled direct channel fext channel direct channel fext channel direct channel measurement fext channel measurement channel gain frequency mhz fig direct crosstalk channel gains various cables using measurement data simulation models loop length higher loop lengths crosstalk channel gain higher direct channel gain diagonal dominance parameter measurement effort directed toward better characterization various empirical models available different cable types vdsl frequencies however extrapolation models frequencies deviate actual measurements moreover system incorporates different topologies deployment using cable types context treatment direct path diagonal channel matrix elements crosstalk paths elements quite different parametric models direct channel developed cable types made available latest recommendation model validated hundreds mhz using results extensive cable measurement campaigns different cable types ongoing effort improve existing models derive models cable types authors proposed model characteristic impedance propagation constant cables using fewer parameters long dsl transmission limited point point technology adsl standard model crosstalk called model requires could chance total actual fext coupling strength real bundle worse obtained parametric models model based model single line crosstalk evaluated worst case sum interferers using parametric model see crosstalk coupling depends frequency coupling length disturber victim attenuation interfering signal vdsl coupling shown proportional frequency similar direct path extrapolation crosstalk channel models vdsl shows fext signal underestimated higher frequencies system extensive measurement campaigns progress standardization crosstalk channel model recently enhanced version worst case deterministic fext model proposed frequencies proposals crosstalk channels shown dual slope high frequencies resulting higher fext signal obtained extrapolation vdsl models model lower slope fext mhz higher slope fext mhz consistent experimental fext data modeling used random nature amplitude variation fext channel much significant direct channel however requires sophisticated crosstalk mitigation modeling sufficient current studies aim better characterize randomness channel terms amplitude phase various stochastic models vdsl adopted distribution magnitude fext channel uniform distribution phase coupling authors also suggested fext different lines binder statistically independent thus modeling direct part quite mature detailed modeling fext randomness currently measurement simulation simulation frequency mhz fig diagonal dominance measure function frequency using measurement simulation data binder lines equal loop lengths vdsl frequencies show strong diagonal dominance whereas channel higher frequency tones diagonally dominant still study simulation results presented herein based separately modeling direct channels fext coupling stochastic parameters fig presents example channel gain fext coupling different cables using stochastic channel models measurement data cable pairs measured seen model higher attenuation model measured cable closer model diagonal channel dominates fext channel vdsl frequencies however fext channel becomes strong higher frequencies diagonal dominance channel matrix focusing single dsl channel matrix contains direct channel coefficients line diagonal elements crosstalk coupling coefficients line elements row channel matrix represents crosstalk paths ulti hannel rosstalk ancellation echniques various crosstalk cancellation methods available literature categorized terms coordination among users binder coordination possible binder behaves like interference channel receiver decodes signal independently presence interference users advantage methods applied independently modem without coordination unfortunately techniques yield relatively low data rate user presence crosstalk shown fig seen data rate without crosstalk cancellation achieved full crosstalk cancellation effect crosstalk reduced using coordinated processing signals downstream dsl matched filter bound single wire performance canceler linear mmse canceler linear canceler approx first order canceler cancellation data rate multiple transmitters single receiver whereas column represents transmission path single transmitter multiple receivers intuitively one would expect signal desired pair much stronger signals coupled pairs shown fig intuition indeed holds true vdsl frequencies mhz fext among pairs insignificant thus vdsl frequencies channel matrix diagonal dominant convenient communication following present short mathematical quantification diagonal dominance property property discussed section addresses fext cancellation distinguish two types diagonal dominant channels downstream transmitting modems colocated receiving modems situated different lengths crosstalk signal disturber must propagate full length victim line interfere victim receiver since insulation twisted pairs increases attenuation diagonal element dominates row thus downstream dsl channel diagonal dominant rwdd channel said rwdd maxi using reciprocity principle dsl channel upstream diagonal dominant cwdd quantified maxi say channel diagonal dominant cwdd rwdd max diagonal dominant channel ensures crosstalk channel matrix hence dsl specific research used feature way another moreover performance analysis various algorithms used metrics bound performance diagonal dominance parameter depicted fig using measurement data simulation results cable length meters binders wires depending scenario channels diagonal dominant less frequency mhz thus diagonal dominance holds distances vdsl frequencies hold many frequencies line length fig average achievable user rate whole bandwidth mhz binder length binder composed users equal line lengths binder behaves like broadcast channel single transmitter generates signals geographically dispersed subscribers enables joint processing transmitter side coordination received signals possible upstream channel mac single receiver receives signals different users see fig joint processing transmit receive side link requires customer premises modems possible case bonded dsl system single typically business customer uses several twisted pairs achieve high rates dsl configurations customer single twisted pair customers situated different locations thus dsl system considered mimo research developed wireless systems applicable well see example references therein however rates dsl systems particular systems high number users served simultaneously system published algorithms wireless communication feasible dsl community turned research low complexity algorithms advent vectored transmission surge research interest transceiver design crosstalk cancellation dsl systems various nearoptimal receivers designed perform crosstalk cancellation relatively low complexity following subsections discuss various crosstalk cancellation schemes upstream downstream transmission crosstalk cancellation upstream starting upstream first discuss theoretical performance bounds present various crosstalk cancellation schemes mac capacity performance bounds capacity mac channel derived cover decades ago capacity characterized achievable rate region described set equations example case rate region given rmfb rmfb det hsh diagonal power matrix matched filter bound given however number users large since practical dsl implementations coding scheme achieve capacity convenient use simple performance bounds leshem zehavi presented efficient rate control mac subject psd mask transmitters system showed need upstream power control long receiver kept sufficiently linear although performance bounds presented achievable quite tight upper bounds practical dsl scenarios hence allow quantify algorithm evaluating close bound demonstrate dsl better schemes close bounds hence close optimal single wire performance swp intuitive approach compare achievable performance case single user transmission single wire pair performance denoted hereafter single wire performance swp case received signal user tested user given hii single user limited additive noise attenuation channel additive noise assumed gaussian distributed assumption gaussian distribution transmit symbols shannon capacity derived using snr expression shannon capacity achievable requires ideal signal processing hence realizable practical systems reason customary dsl systems model system imperfections single parameter commonly referred shannon gap shannon gap includes types imperfections starting amplifier noise ending use square qam constellation instead theoretical gaussian shaping thus target probability error snr gap taken dsl systems single tone width swp user given rswp transmit signal power noise variance however worth noting swp upper bound achievable performance although fext typically degrades performance instances transmission different modems combined coherently crosstalk wires hence following subsection presents useful upper bound user rate matched filter bound mfb capacity achieved single user utilizes direct well fext coupling reception commonly termed matched filter bound mfb also known single user bound sub dsl terminology modems receive single users received signal signal becomes column channel matrix optimal processing received signal case known matched filter maximal ratio combining mrc receiver simply requires linear combining received signals using thus achievable capacity user rmfb note power bound power allowed single user power allowed network practical scenario simply used bound performance case vdsl system marginal difference mfb swp performance since crosstalk channels much smaller direct channel gains see subsection iiib details since crosstalk increases frequency already presents notable gap swp mfb crosstalk cancelers case mac capacity achieved detecting single user time subtracting detected signal received signal continuing detect next user scheme known successive interference cancellation sic generalized dfe gdfe noted dsl systems noise margin means error propagation unlikely optimality sic requires linear optimal detection stage using mmse receiver subtraction signal successful decoding error correction code well proper ordering users requirements incur significant complexity following present simplified version nearoptimal dsl systems consider decision feedback equalizer dfe receiver based decomposition channel result rates close mfb computation decomposition matrix yields unitary matrix upper triangular matrix first unitary nature matrix made use linear operation received vector get rotated version received vector noise gaussian independent identically distributed lines multiplication unitary structure matrix change statistics noise upper triangular nature matrix made use symbols estimated starting last row cancellation operation signal given dfe sensitive error propagation error decision based increases probability error subsequent user detection however proper matching modulation signal noise ratio error probability made small enough error propagation negligible case spectral efficiency user expressed rdfe obviously user performance significantly affected user ordering user detected first later detailed discussion various user ordering schemes see linear crosstalk cancelers reduce complexity straightforward approach use linear crosstalk cancelers use linear receivers attracted research interest fext cancellation several variants linear receivers example based criterion zero forcing minimum mean square error mmse structures much simpler due absence feedback operations zero forcing canceler implied name attempts cancel assuming disturbance present thus ignoring even awgn kind interference application inverse operator fzf receiver application linear canceler received signal cancels crosstalk upstream dsl system user receives inv signal hinv row seen processing amplifies power additive noise resultant noise pinterference power becomes thus resulting performance strongly dependent condition number channel matrix becomes poor matrix close singular linear mmse canceler minimizes mean square error mse output canceler true value argminf fyk given fmmse results spectral efficiency rmmse implementation complexity mmse almost identical generally better solution use mmse requires knowledge noise covariance matrix however matrix still yield better performance nevertheless advantage mmse negligible channels well snr high thus mmse performs significantly better high frequency tones linear cancelers upstream dsl system illustrate operation linear crosstalk cancelers consider simple case users users transmit equal power twisted pairs equal length experience equal power additive noise direct channel gain users crosstalk coupling thus simple case hence diagonal dominance parameter transmit received signal vectors expressed without crosstalk canceling crosstalk signal interferes reception desired signal received presence noise term power interference term power decreases rate thus spectral efficiency snrawgn rfext snrawgn snrawgn purposes comparison note single wire performance rswp snrawgn whereas matched filter bound even higher rmfb snrawgn thus high snr even small values cause significant performance degradation crosstalk canceler uses inverse channel matrix receiver preprocessing fzf thus estimate symbol user seen processing removes effect crosstalk enhances noise folding noise users spectral efficiency canceler user given rzf snrawgn contrast cancellation case canceler performs well whenever approaches swp performance hand canceler performs poorly channel matrix close singular mmse canceler simple matrix results spectral efficiency mmse snrawgn snrawgn comparing performance mmse coincides high snr however signal experiences situations attenuation large external interference dominates receiver noise lower awgn snr mmse canceler performs better canceler minor increase complexity spectral efficiency awgn snr matched filter bound single wire performance linear mmse canceler linear canceler awgn snr diagonal dominance parameter fig performance linear cancelers versus parameter two users awgn snr low crosstalk coupling mfb close swp swp metric sufficient large mfb becomes relevant metric performance comparison figure shows snr higher mmse practically advantage except case channel matrix close singular lower snrs advantage mmse greater mmse canceler performs better precoding stated optimal nonlinear processing asymptotically achieve sum rate capacity downstream using dirty paper coding multidimensional lattice precoding however approaches require high implementation complexity hence considered dsl simpler scheme considered dsl precoder viewed lattice precoding recalling fext modeled channel matrix thp cancels interference using decomposition conjugate transpose channel matrix given unitary matrix upper triangular matrix precoding operation divided two parts modulated signal first undergoes interference using elements modulo operation rotated cancel channel rotation using matrix remove interference associated previous users symbol user precoding operation symbol evaluates lower increase required power symbol undergoes modulo operation crosstalk cancellation downstream downstream crosstalk transmission signals downstream processing attempts cancel crosstalk also another consideration deal precoded signals line operate within assigned power spectral mask another important difference upstream lack channel estimation central office section assume perfect channel precoding deal cancellation performance various precoders notes channel estimation given section capacity capacity region gaussian derived many authors work based concept dirty paper coding makes possible transmit data without degradation presence interference known transmitter equivalently capacity also achieved using lattice precoding discussed erez derivation gaussian capacity beyond scope review paper hence upstream focus single wire performance swp matched filter bound mfb single wire performance downstream similarly derived upstream mfb downstream transmission capacity modems transmit single user mfb user rdown row downstream channel matrix allowed transmission power line considered tone case bound achieved dirty paper coding mod modulo operation defined result within square edge centered origin complex plane collecting symbols resulting vector rotated applying given used decomposition matrix hermitian resulting received signal comparing effect channel operation seen interference users eliminated step left normalize signal reciprocate modulo operation additional modulo operation receivers thus estimated symbol user mod mod thanks modulo operations thp cancel interference almost cost contrasts linear precoding schemes see next also remove interference power cost significant price modulo operation comes effective change channel gaussian signaling longer optimal loss called shaping loss moreover although gaussian signaling maximizes achievable rate dsl schemes limited square qam modulations hence lose shaping loss regardless interference cancellation method thus actual loss modulo operation negligible precoder thp thp efficient simple method remove interference transmitted symbols almost cost equations describe thp may somewhat intimidating principle operation quite simple following illustrate operation thp simple case users consider downstream transmission two users simplicity consider real valued channel matrix decomposition transpose let assume symbols transmission users using symbols precoded ignoring modulo operation symbol vector rotated transmitted vector recalling matrix unitary qqt thus thus get hence symbols received two separate receivers without interference schematic diagram presented fig however description ignored power requirement transmission particular assuming symbols statistically independent equal power transmission requires power power values result extreme waste power avoid power loss use modulo operation assume transmit symbols lie range thus instead using use integer chosen resulting processed symbol also range receiver use knowledge range transmission values reconstruct estimate constructed easily found requiring symbol inside allowed range cost modulo operation slightly lower power efficiency typically negligible compared reduction interference modulo detector modulo detector modulo modulo modulo fig succesive using factorization modulo operation simple channel linear precoders equivalent mmse canceller downstream typically termed diagonal loading precoder signal leakage ratio slnr precoder upstream precoder typically perform slightly better precoder using almost implementation complexity however precoder studied extensively context dsl hence discussed detail linear precoder simple technique precompensates true symbol vector inverse channel matrix fzf diag precoded signal vector becomes fzf diagonal scaling matrix chosen total transmit power line satisfies power mask constraint tightly therefore element gii different transmit power scaling linear precoder illustrate transmit precoding necessity gain scaling consider downstream transmission two users channel matrix linear precoder without gain scaling fzf diag fzf signals precoding becomes users transmit maximum power mask transmit powers precoding users amplified factor easy see precoded signals violate psd mask proper scaling precoded signals required transmission elements diagonal scaling matrix computed argmaxi diag row scaling precoded signal received signals free crosstalk incur power penalty algorithm mmse legacy techniques vectored vdsl system extraordinary success vectoring techniques vdsl systems proved pivotal moving toward next generation standard however techniques developed vdsl vectoring relied diagonal dominance channel enable low enough implementation complexity discussed subsection diagonal dominance hold frequency range nevertheless subsection briefly discuss methods better depict challenges transition vdsl crosstalk cancellation vdsl works showed based linear yields performance close single user bound contrast wireless mimo possible bound performance canceler using diagonal dominance parameter thus vdsl systems derived using analytic estimates vectoring gain without needing rely results analysis based cancelers improved bounds prove strongly matrices noise enhancement upstream power penalty downstream negligible performance close optimal shown data rate achievable user upstream dsl system bounded rzf max thus lower frequencies property significant significantly smaller depicted fig canceler quite close upper bound hence hand bounds meaningless hence guarantee performance canceler similar performance found downstream system interestingly shown fig although bounds become meaningless performs quite well well thus many scenarios still use use check performance point important note complexity complex time implementation vdsl systems required another step using property following implementation vdsl systems relied performance azf complexity seen negligible linear precoder achieves performance however degradation performance channel deviates diagonally dominant table summary crosstalk cancellation techniques data rate spectral efficiency users snrawgn comments used downstream negligible power penalty due modulo operation requires ordering optimize performance used upstream requires ordering optimize performance problem error propagation feedback loop performs marginally better requires knowledge noise covariance requires channel inversion lower frequencies require channel inversion longer loops lower frequencies suitable typical applications low complexity approximation shown good small enough fig shows performance using first oder approximation equivalent techniques approximation described following subsection approximate canceler express channel matrix matrix containing elements power series expansion convergent eigenvalues matrix less one leshem used property channel show first order approximation fazf achieves capacity complexity reduction significant matrix inverse operation replaced inverse diagonal matrix requires single element inversions computed anyhow using feq frequency domain equalizer coefficients important since requires channel matrix inversion tone matrix inversions frequently required due changes user status variations crosstalk characteristics hence cause increase overhead computational cost receiver unfortunately approximation turned useful small hence carry well systems see fig avoid channel matrix inversion iterative algorithms crosstalk cancellation dsl systems proposed however schemes perform well higher frequencies systems introduce latency systems need adopt complicated algorithms whose design considerations detailed section crosstalk cancellation using approximate inversion illustrate approximate inversion method extend model box three user model denote diagonal matrix matrix containing elements first order approximation method approximates inversion canceler fazf canceler remove crosstalk completely since fazf identity applying canceler fazf received signal results signal estimate spectral efficiency razf snrawgn snrawgn thus approximate close optimal snrawgn much better cancellation case snrawgn however small snr high frequencies performance approximate significantly lower exact spectral efficiancy awgn snr single wire performance linear canceler approx inversion method cancellation awgn snr diagonal dominance parameter fig performance linear approximate inversion canceler versus parameter three users awgn snr parameter typical vdsl linear canceler performs close awgn performance adaptive crosstalk cancellation adaptation crosstalk cancellation schemes requires update canceler coefficients based dynamics channel upstream vdsl several adaptive algorithms minimize mean square error mse output crosstalk canceler certainly popular algorithm lms algorithm lms stochastic gradient descent method certain conditions converges mmse solution popularity lms algorithm mostly due simplicity lms crosstalk canceler tone described matrix denotes canceler vector line time hence output lms crosstalk canceler written error vector lms recursion written matrix form denote value time precoding matrix received symbol modulated data respectively appropriate selection step size lms guaranteed converge nevertheless convergence rate depends eigenvalues input correlation matrix thus lms based adaptive algorithms efficient converge quite rapidly diagonally dominant vdsl channels hand algorithms become inefficient converge slowly higher frequencies due bad conditioning input correlation matrix hence conventional lms algorithm suitable system new methods needed vdsl system adaptation crosstalk canceler matrix downstream complicated transmitter optical network unit onu directly measure channel matrix hence calculation precoding matrix must rely feedback receivers two main types feedback considered channel estimation feedback signal error feedback channel estimation feedback based transmission orthogonal synchronized pilot symbols transmitters simultaneously estimation row channel matrix receiver estimated row transmitted upstream back transmitter uses construct full channel matrix calculate precoding matrix signal error feedback method based feedback quantized version error signal measured receiver onu groups error signal feedback cpe vector adapt precoder coefficient matrix simpler version adaptive precoder proposed independently louveaux van der veen bergel leshem adaptive precoder shown robust converged channels strong rwdd property however approaches lost attractiveness new standard adopted tdd approach thus onu directly measure channel upstream phase use estimation downstream phase need feedback esign onsiderations hallenges fast transmission technology development different features current vectored vdsl technology techniques developed vdsl system thus simply applied system yet efficient design methodologies system required deliver data rate gbps per user short telephone lines recently various novel techniques developed system techniques still sufficient reach desired rates currently available hardware section discuss various differences vdsl give overview recent research highlight design considerations system also point topics important implementation studied sufficiently require research characteristics first present concise representation main characteristics complement general system model given section transmission model consists vectored users typically upto operates loop lengths shorter standard targets aggregate data rate combined upstream downstream gbps per user data rate performance system limited fext since next eliminated tdd scheme provides independent transmissions upstream downstream directions whole bandwidth transmission bandwidth starts mhz ends mhz low bandwidth version mhz upcoming version system employs dmt modulation size width khz user corresponding total mhz mhz system channel matrix lower frequencies higher frequency tones user transmits qam symbol rate symbols per second unit energy gain scaling factor control transmission power psd mask based frequency operation mhz mhz mhz mhz transceiver total maximum transmit power dbm additive noise awgn psd target ber set snr gap specified noise margin coding gain lead transmission gap limits bit loading bits per tone qam constellation size points channel coherence time dsl channel slowly since consists copper wires static cable binder fixed user terminals time variability attributed mainly changes customer wiring temperature variations time scale minutes therefore dsl channel long coherence time defined time effective variation channel impulse response several advantages regards dsl system design contrast wireless communication systems furthermore allows operation many users simultaneously order magnitude larger existing mimo wireless systems long coherence time allows almost perfect csi acquisition enables robust design crosstalk cancellation schemes addition tracking updating channel matrix less frequent reduces overhead required pilot symbols inserted data symbols large channel coherence iterative algorithms power allocation well crosstalk cancellation feasible even large number lines channel estimation calibration discussed long coherence time enables receivers obtain good channel estimates important cross talk cancellation hence channel matrix estimation upstream transmission much easier downstream upstream transmission users transmit known training symbols simple least squares technique used estimate channel matrix distribution point length user training symbols least equal number vectored users downstream channel estimation carried user terminals row channel matrix estimated increases complexity users modem power consumption however main challenge forward estimated channel user distribution point vdsl systems based fdd scheme channel coefficients quantized user transmitted upstream channel increases overhead complicates design reduction quality csi transmit precoding tdd duplexing scheme makes possible exploit channel reciprocity avoid complicated feedback protocol also enables perform csi related tasks using channel reciprocity downstream channel estimated estimated upstream channel practice transmit receive paths identical hence estimated downstream channel needs calibrated compensate mismatch calibration process still development yet received sufficient academic attention adaptive crosstalk cancellation adaptation crosstalk canceler matrix track channel dynamics another important design consideration system general traditional adaptive algorithms designed vdsl system applied system however algorithms mainly lms algorithm become inefficient converge slowly higher frequencies due bad conditioning input correlation matrix higher frequencies received signal covariance matrix badly conditioned due loss diagonal fig lms algorithm adaptive crosstalk cancellation upstream dominance characteristics ongoing research adaptive techniques upstream transmissions interesting new paper presents novel algorithm speed convergence lms crosstalk canceler preprocessing input received signal judicially designed matrix preprocessing matrix shown fig proposed lms algorithm first preprocesses input signal multiplying matrix applies standard lms preprocessed signal updates another decoding matrix conventional lms accelerated updating carefully selected times preprocessing matrix include also lms matrix lms paper shows updates preprocessing matrix convergence lms crosstalk canceler reducing condition number correlation matrix lms input since preprocessing matrix frequently updated complexity algorithm approximately twice complexity conventional lms downstream tdd scheme facilitates simpler approach adaptive algorithms vdsl using channel reciprocity feedback users longer necessary instead adaptive algorithm carried solely upstream resulting fext cancellation matrix guaranteed also good precoding matrix downlink optimized crosstalk cancellation schemes performance crosstalk cancellation techniques described section depends mainly characteristics dsl channel since channel diagonally dominant higher frequencies crosstalk cancellation schemes vdsl system longer optimal channel bandwidth overview recent research works describe novel approaches deal strong crosstalk system power normalization use linear cancellation downstream transmission requires energy normalization guarantee power constraint lines vdsl low frequencies shown simple power normalization results near optimal performance case frequencies power normalization significantly reduce snr channel matrixp single normalization factor maxi gain scaling shown near optimal low frequencies simple power normalization approach adopted two stage normalization first stage power user normalized according overall line power second stage simple overall power normalization vdsl effective approach optimizes power normalization scheme via linear program improve performance balances rates among users ordered successive interference cancellation interference cancellation schemes downstream require power normalization power increase negligible modulo operation thp however issue user ordering successive interference becomes important channel gains depend decoding order users precoded earlier experience higher data rates users precoded later achieves lower data rates creating undesirable rate variance among users several works suggested novel approaches user ordering improve total throughput reduce rate variance upstream ordering affects performance strongly depends detection ordering due error propagation effect comparing wireless communications requires complicated processing simultaneously serves user higher user rates hand low rate change channel user activity allows longer processing time optimization problem problem becomes even complicated take whole bandwidth account try jointly optimize complete network discussed following subsection joint crosstalk cancellation dsm dynamic spectrum management dsm techniques shown improve performance dsl systems improving resource allocation users system bandwidth dsm evolved mostly independently multiuser crosstalk cancellation considered separately different level coordination dsl systems novel research approaches consider joint signal spectrum coordination joint implementation crosstalk cancellation spectrum management formulated optimization problem need derive optimal precoder decoder optimal power allocation subject given set constraints practical systems typically face three power constraints first constraint psd mask mask represents maximum psd allowed tone constraint defined regulation prevents allocating much power certain frequency band second constraint transmit power diag pline bcap fig average achievable user rate whole bandwidth mhz binder length binder composed users equal line lengths spectral efficiency line length diag mask matched filter bound single wire performance canceler linear mmse canceler linear canceler approx first order canceler cancellation data rate pline account limited range transmit amplifiers third constraint bit cap bcap reflects limited capabilities modulator allow constellation sizes greater threshold noted similar optimization problems studied extensively wireless networks however proposed algorithms wireless systems directly applicable context due use large number users dsl systems well additional constraint power pline addition psd mask mask existing power constraint joint optimization precoder powers obtained example max matched filter bound single wire performance canceler linear mmse canceler linear canceler approx first order canceler cancellation achievable rate line tone given set power allocation vectors precoding matrix desired ence cancellation scheme vectors mask pline denotes psd masks powers frequency mhz users tone respectively case mmse precoding solved using closed form solution precoders given power allocation fig spectral efficiency performance using channel meaper tone per line using duality surement binder lines length recently mmse precoding proposed nonlinear thp framework algorithms solve optimization problems quite complex systems well stochastic mimo channel models however techniques useful provide cables measured channel contains data putable bound compare computationally efficient algorithms cable pairs measuring plot consider simulated channel two scenarios optimization problems reduce complexity algorithms dsm first scenario consists lines equal length rithms applied fixed canceler structure range second scenario considers transmit power optimization minimizes residual lines different line lengths uniformly spaced crosstalk improves performance shown intervals lowest frequency mhz application linear canceler mhz mhz systems decouples power allocation problem essentially defined standard backwards compatibility comes independent user scenario iterative legacy adsl systems noise psd water filling iwf algorithm applied user transmit psd specification tone taken maximize data rate selfishly results bit cap bits per tone qam optimal performance gives low complexity solution densest constellation maintained effective snr readily implemented current research aims find shannon gap used standard recommends linear crosstalk cancelbetter algorithms fill gap infeasible optimal ers mhz since loss linear scheme solution simplified solution marginal mhz see fig fig shows linear crosstalk schemes near design consideration crosstalk cancellation schemes optimal mhz note high frequency tones important design issue crosstalk mhz diagonal dominance strong cancellation scheme discussed section degrades performance linear schemes shown tradeoff linear cancelers fig however high frequencies spectral simpler lose performance discuss choice efficiency significantly lower due signal attenuation crosstalk cancelers regards performance standard therefore overall loss data rate computed channels performance evaluated using measured channel tones marginal performance linear approximate data rate per user sec matched filter bound single wire performance canceler linear mmse canceler linear canceler approx first order canceler cancellation line length fig achievable data rates users binder uniformly spaced line lengths users use fast mhz various cancelers inversion significantly degraded recommended seen linear mmse canceler performs marginally better higher frequencies however recommended since provides extra protection external noise complexity almost identical recently asymptotic analysis performance receiver using theory large dimensional random matrices presented system channel mhz diagonally dominant linear scheme longer crosstalk cancellation see fig cancellation schemes preferred mmse linear scheme still gives reasonable performance degradation compared mfb compared schemes computationally complex linear techniques however performance improved proper ordering procedures among users recent works show impact channel estimation error crucial system vdsl system also important design consideration system noted simulation results obtained without optimal power control discussed subsection algorithms joint dsm crosstalk cancellation showed significant performance improvement optimized linear higher frequencies however considered algorithms higher computational complexity due power allocation systems implementation complexity crosstalk cancellation schemes require significant computational resources given various signal processing operations large scale first memory requirement processor store channel coefficients per tone since number tones large memory requirement significant efficient usage memory task also needs investigation example memory requirement reduced channel coefficients stored tones throughout band interpolated tones however reduce data rate performance error channel estimates increases interpolation interval tradeoffs standard signal processing design loss due quantization interpolation characterized using techniques another complexity consideration computation matrix linear mmse require inversion many channel matrices whose complexity large decomposition thp precoder requires similar high computational resources approximate inversion method avoids matrix inversion recommended system higher frequency tones adaptation coefficients simple need costly computation standard provisions single sync symbol every data symbols track channel authors proposed scheme directly track canceler matrix whose convergence depends diagonal dominance parameter channel research required derive computationally efficient adaptive canceler schemes channel however major complexity occurs matrix applied vector size requires instructions per second becomes prohibitively high gips billion instructions per second users binder operating khz tones deployment coexistence penetration fibers closer end users various deployment scenarios considered fiber distribution point fttdp system fttdp consists scenarios either mounted pole placed underground curb building basement etc fttdp fiber node far serve dozen users close serve single user since scenario requires deployment many new dps also considers possibility powered customer cpe technology known reverse powering eliminates need power infrastructure hence lowers deployment cost technology also introduces new challenges use single wire broadband power transmissions issues related safety guidelines power transmission home network another issue affecting deployment overlap bandwidth radio broadcast services radio services overlap consecutive tones system throughout band tones need either masked notched interference experienced radio service broadcast radios broadband services cover large bandwidth mhz system avoid data transmission bands limits rate performance note unused tones employed system provisioning emphasized efficient transmissions methods used enable system broadband technologies without disturbing band profile coexistence system legacy lines another important issue vectored vdsl systems already deployed various service providers may coexist users binder vdsl fall back coexistence necessary mass deployment service providers require migration vdsl seamless must able peacefully common approach technologies use nonoverlapping spectra simple approach avoid interference two technologies however leaving lower frequency bands spectral efficiency high limits data rate performance technology coexistence vdsl overlapping spectrum challenging problem somewhat similar problems faced adsl deployment japan parallel tdd based hdsl crosstalk cancellation discontinuous operation dsl access networks boast always distinguishing feature broadband service however typical dslam fully transmit data time yet dsl lines switch exploiting modes highly effective creating green access network moreover energy efficiency important local power supply rather reverse powered subscribers via copper wires discontinuous operation basically means data symbols transmitted data available vdsl system sends idle data packets system mute data symbols switch analog components enhance energy efficiency crosstalk cancellation conjunction discontinuous operation become challenging concise problem formulation precoded discontinuous process presented crosstalk cancellation must maintained active lines lines discontinuous mode novel algorithms required adapt linear precoder corresponding active lines order enjoy energy savings well crosstalk cancellation iscussion uture irections article provided overview multiuser signal processing techniques crosstalk cancellation next generation system discussed salient features upcoming dsl technology highlighted key differences predecessors tutorial article shows considerable advances made recent years crosstalk cancellation vectored dsl systems however system poses new challenging problems research still needed fully realize goal achieving gigabit date rates telephone channels access networks use wider bandwidth tdd duplexing scheme among others two distinguishing features require special attention signal processing community dsl channel many specific characteristics differ wireless channels unique characteristics allowed implementation multiuser interference cancellation possible large number users channel still integrates characteristics loses key one higher frequencies longer diagonal dominant showed approximate one enabler massive vectoring perform well enough frequencies provided overview important techniques linear nonlinear explained simple examples however techniques require high implementation complexity research required make multiuser processing work dsl channel mhz similarly showed adaptive schemes designed vdsl system perform well channel additional research required address convergence algorithms addition efficient calibration processes use channel reciprocity downstream transmission additional issues require research include study mitigation techniques interference uncoordinated lines noise external binder vectored system dynamic spectrum management better channel modeling far channel modeling higher frequency less accurate vdsl measurement campaigns additional statistical studies required improve modeling furthermore characterization crosstalk channel early steps publications addressed heavily rely crosstalk cancellation good stochastic modeling required developing testing algorithms use system alternative backhauling option networks could keep copper gold category eferences starr cioffi silverman understanding digital subscribe line technology upper saddle river timmers guenach nuzman maes evolving copper access network ieee communications magazine vol august recommendation draft recommendation fast access subscriber terminals fast physical layer specification oksman strobel wang wei verbin goodson sorbara new standard brings dsl gigabit era ieee communications magazine vol march high speed digital subscriber line recommendation series transmission systems media digital systems asymmetric digital subscriber line adsl recommendation series transmission systems media digital systems cioffi mosheni leshem gdsl gigabit dsl contribution cioffi jagannathan mohseni ginis cupon copper alternative pon dsl networks ieee communications magazine vol june ginis cioffi vectored transmission digital subscriber line systems ieee journal selected areas communications vol june cendrillon ginis van den bogaert moonen nearoptimal linear crosstalk canceler upstream vdsl ieee transactions signal processing vol linear crosstalk precoder downstream vdsl ieee transactions communications vol may leshem low complexity linear precoding technique next generation vdsl downstream transmission copper ieee transactions signal processing vol louveaux van der veen adaptive precoding downstream crosstalk precancelation dsl systems using feedback ieee transactions signal processing vol june bergel leshem convergence analysis downstream vdsl adaptive multichannel partial fext cancellation ieee transactions communications vol october shannon mathematical theory communication bell system technical journal vol july bingham multicarrier modulation data transmission idea whose time come ieee communications magazine vol lin statistical behaviour multipair crosstalk dominant components bell systems technical journal vol werner hdsl environment ieee journal selected areas communications vol parametric modeling twisted pair cables vdsl ansi contribution jan van den brink tno cable reference models simulating metallic access networks etsi stc permanent document june chen dsl simulation techniques standards development digital subscriber lines indianapolis usa macmillan karipidis sidiropoulos leshem experimental evaluation capacity statistics short vdsl loops ieee transactions communications vol july karipidis sidiropoulos leshem tarafi ouzzif crosstalk models short lines measured data eurasip journal applied signal processing vol van den brink van den heuvel crosstalk twisted pair cabling measurements modelling contribution van den brink dual slope behaviour contribution acatauassu host berg klautau borjesson simple causal copper cable model suitable frequencies ieee transactions communications vol nov humphrey release cable measurements use simulations contribution january van den brink modeling behavior elfext twisted pair quad cables ieee transactions communications vol sorbara duvaut shmulyian singh mahadevan construction channel model evaluation fext cancellation systems ieee sarnoff symposium april maes guenach peeters statistical mimo channel model gain quantification dsl crosstalk mitigation techniques international conference communications icc dresden germany june schroeder hoeher stochastic mimo model crosstalk vdsl cable binders ieee international conference communications icc dresden germany june ginis peng alien crosstalk cancellation multipair digital subscriber line systems eurasip app signal vol article pages yang hanzo fifty years mimo detection road mimos ieee communications surveys tutorials vol fourth quarter paulraj mimo wireless linear precoding ieee signal processing magazine vol sept gesbert kountouris heath chae salzer shifting mimo paradigm ieee signal processing magazine vol sept rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays ieee signal processing magazine vol jan cover thomas elements information theoryy new york wiley leshem zehavi rate control psd limited multiple access systems linear programming ieee international conference acoustics speech signal processing icassp may chen seong zhang cioffi optimized resource allocation upstream vectored dsl systems generalized decision feedback equalizer ieee journal selected topics signal processing vol verdu multiuser detection cambridge university press new york cioffi sum capacity gaussian vector broadcast channels ieee transactions information theory vol sept viswanath tse sum capacity vector gaussian broadcast channel duality ieee transactions information theory vol aug caire shamai achievable throughput multiantenna gaussian broadcast channel ieee transactions information theory vol july vishwanath jindal goldsmith duality achievable rates capacity gaussian mimo broadcast channels ieee transactions information theory vol oct weingarten steinberg shamai capacity region gaussian broadcast channel ieee transactions information theory vol sept zamir shamai erez nested codes structured multiterminal binning ieee transactions information theory vol jun patcharamaneepakorn armour doufexi equivalence slnr mmse precoding schemes singleantenna receivers ieee communications letters vol july maes nuzman energy efficient discontinuous operation vectored ieee international conference communications icc june neckebroek moeneclaey coomans guenach tsiaflakis moraes maes novel bitloading algorithms coded dsl transmission linear nonlinear precoding ieee international conference communications icc june leung huberman vectored dsl potential implementation issues challenges communications surveys tutorials ieee vol bergel leshem signal processing vectored multichannel vdsl academic press library signal processing communications radar signal processing vol zafaruddin prakriya prasad performance analysis zero forcing crosstalk canceler vectored ieee signal processing letters vol april bergel leshem performance zero forcing dsl systems ieee signal processing letters vol may dai poor turbo multiuser detection coded dmt vdsl systems ieee journal selected areas communications vol feb zafaruddin prakriya prasad iterative receiver based sage algorithm crosstalk cancellation upstream vectored vdsl isrn communications networking hindawi vol article pages wahibi ouzzif masson saoudi crosstalk cancellation upstream coordinated dsl using iterative mmse receiver ieee international conference communications june widrow hoff adaptive switching circuits louveaux van der veen adaptive dsl crosstalk precancellation design using feedback end users ieee signal processing letters vol binyamini bergel leshem arbitrary partial fext cancellation adaptive precoding multichannel downstream vdsl ieee transactions signal processing vol binyamini bergel adaptive precoder using sign error feedback fext cancellation multichannel downstream vdsl ieee transactions signal processing vol may duvaut biyani mahadevan singh maheshwari adaptive mimo canceller odmc upstream dsl self fext cancellation proceedings european signal processing conference eusicpco laussane switzerland zanko bergel leshem gigabit dsl approach european signal processing conference eusipco gigabit transmission unshielded twisted pair cables arxiv preprint muller eriksson host klautau optimizing power normalization linear precoder linear programming ieee international conference communications icc june muller eriksson klautau rate balancing based tomlinson harashima precoding systems ieee communications letters vol aug hekrdla matera wang wei spagnolini ordered precoding downstream ieee global communications conference globecom dec strobel joham utschick achievable rates implementation limitations hybrid networks ieee international conference communications icc june strobel barthelme utschick mmse precoding ieee global communications conference globecom dec barthelme strobel joham utschick weighted mmse precoding ieee global communications conference globecom dec ginis cioffi distributed multiuser power control digital subscriber lines ieee journal selected areas communications vol jun cendrillon moonen verlinden bostoen optimal multiuser spectrum balancing digital subscriber lines ieee transactions communications vol may tsiaflakis diehl moonen distributed spectrum management algorithms multiuser dsl networks ieee transactions signal processing vol oct huberman leung dynamic spectrum management dsm algorithms xdsl ieee communications surveys tutorials vol first boccardi huang precoding mimo broadcast channel power constraints ieee workshop signal processing advances wireless communications july lan transmitter optimization downlink power constraints ieee transactions signal processing vol june shi schubert boche power constrained rate optimization multiuser mimo systems international itg workshop smart antennas feb christensen agarwal carvalho cioffi weighted maximization using weighted mmse mimobc beamforming design ieee transactions wireless communications vol december recommendation fast access subscriber terminals power spectral density specification zafaruddin klein bergel leshem asymptotic performance analysis zero forcing dsl systems ieee international conference science electrical engineering icsee nov maes nuzman tsiaflakis sensitivity nonlinear precoding imperfect channel state information european signal processing conference eusipco march sayag leshem finite word length effects transmission rate zero forcing linear precoding multichannel dsl ieee transactions signal processing vol medeiros magesacher eriksson odling vectoring may cause neighborhood wars ieee international conference communications icc june annex symmetric digital subscriberline adsl transceivers recommendation series transmission systems media digital systems strobel utschick discontinuous operation precoded ieee international conference acoustics speech signal processing icassp march zafaruddin prakriya prasad performance linear energy receiver self alien crosstalk mitigation upstream vectored digital subscriber line iet communications vol zafaruddin pierrugues performance dual sensor based interference cancellation scheme downstream dsl ieee international conference communications icc may authors iographies zafaruddin smzafar received degree jamia millia islamia university new delhi india degree kurukshetra university kurukshetra india electronics communication engineering degree electrical engineering indian institute technology delhi india phd research crosstalk cancellation vectored dsl systems cto office red bank usa ikanos communications india pvt acquired qualcomm bangalore india involved research development xdsl systems since faculty engineering university ramat gan israel researcher working signal processing wireline wireless communications awardee pbc vatat fellowship program outstanding researchers china india council higher education israel two years current research interests include signal processing communications communication protocols multichannel communication cellular networks sensor networks xdsl systems itsik bergel received electrical engineering magna cum laude physics magna cum laude ben gurion university israel respectively summa cum laude electrical engineering university tel aviv israel respectively senior researcher intel communications research lab received yitzhak chaya weinstein study award postdoctoral researcher dipartimento elettronica politecnico torino italy working capacity noncoherent channels currently faculty member faculty engineering university israel main research interests include multichannel interference mitigation wireline wireless communications cooperative transmission cellular networks cross layer optimization random networks since bergel serves associate editor ieee transactions signal processing amir leshem leshema received cum laude mathematics physics cum laude mathematics degree mathematics hebrew university jerusalem israel respectively faculty information technology systems delft university technology netherlands postdoctoral fellow working algorithms reduction terrestrial electromagnetic interference antenna arrays signal processing communication director advanced technologies metalink broadband responsible research development new dsl wireless mimo modem technologies served member several international standard setting groups etsi ieee also visiting researcher delft university technology joined university one founders faculty engineering full professor head communications research track technical manager consortium developing technologies provide mbps beyond copper lines spent sabbatical delft university technology stanford university main research interests include multichannel wireless wireline communication applications game theory dynamic adaptive spectrum management communication networks array statistical signal processing applications multiple element sensor arrays networks wireless communications imaging set theory logic foundations mathematics leshem associate editor ieee transactions signal processing leading guest editor special issues signal processing astronomy cosmology ieee signal processing magazine ieee journal selected topics signal processing
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flows lattice boltzmann simulation floating bodies simon bognera ulrich jan lehrstuhl systemsimulation erlangen abstract paper presents model simulation flows means lattice boltzmann method approach built upon previous works simulation particle suspensions one hand free surface model show two approaches unified novel set dynamic cell conversion rules evaluation concentrate rotational stability rigid bodies floating plane water surface classical hydrostatic problem known naval architecture show consistency method kind flows obtain convergence towards ideal solution measured heeling stability floating box introduction since establishment lattice boltzmann method lbm become popular alternative field complex flow simulations application particle suspensions propelled significant part works ladd aidun based approach momentum exchange method possible calculate hydromechanical stresses surface fully resolved solid particles directly lattice boltzmann boundary treatment paper aforementioned fluidsolid coupling approach extended free surface flows problem solid bodies moving freely within flow two immiscible fluids use free surface model simulate liquid phase interaction gas means volume fluid approach special kinematic free surface boundary condition interface two phases assumed sharp enough modeled locally defined boundary layer boundary layer updated dynamically according liquid advection set cell conversion rules paper proposes unification update rules free surface model particulate flow model also requires dynamical mapping respective solid boundaries lattice boltzmann grid described resulting scheme allows full freedom motion solid bodies flow calculated according rigid body physics demonstrate consistency combined method means simple advection test floating body stratified channel flow discuss main source error dynamic boundary handling particles motion preprint submitted elsevier january figure stencil weights apply method problem rotational stability rigid floating structures kind hydromechanical problems typically emerge marine engineering floating stability offshore structures concern stability ship heeled position static nature addressed problems numerical issues arising effects widely discarded makes wellsuited verification involved force calculations addition provide possibility check convergence simulated systems state equilibrium succeeded showing basic convergence given provided adequate spatial resolutions chosen special problem floating stability cuboid structures convergence numerical simulations towards analytical model obtained idea evaluating simulated floating stability rigid bodies inspired proposed test case based simulator originally developed estimation green water loads ship decks mentioned lattice boltzmann based interaction techniques developed simulation particulate flows able find publications discussing application free surface flows interaction floating structures knowledge approach handle similiar problems one proposed method isothermal lattice bgk method assume lattice model flows set discrete lattice velocities theoretical considerations section however often fall back implicitly native model simplifies explanations figures lattice velocities also called lattice directions lattice links respective weights shown fig following used denote index lattice velocity let positive real numbers divisible spatial resolution constant domain divided cells cubic volumes length yields computation domain discrete lattice cells spatial quantities like commonly given certain unit length metres however dealing lbm specific computations dimensionless lattice coordinates used spatial coordinates thus given following multiples speaking cell positive integer numbers mean lattice cell respective volume lattice refer point cell center lattice direction name particle distribution function pdf direction cell time step lattice bgk propagation scheme derived classic boltzmann equation collision operator substituted bhatnagar gross krook bgk operator including external force term lattice bgk lbgk equation reads feq dimensionless relaxation time related kinematic viscosity lattice speed sound constant model equilibrium function therewith given low mach number expansion maxwell distribution function feq valid small flow velocities following constraint holds external force term used represent gravitation expressed acceleration simulation given local macroscopic quantities density fluid momentum obtained moments pdfs last equation contains momentum flux tensor stress tensor represents shear stresses flow system equlibrium feq stress tensor vanishes within lattice boltzmann model pressure linearly related density equation simulation includes external force order simulate effect gravity stability concerns limit density gradient therefore hydrostatic pressure gradient buick greated give following incompressibility condition relates external force projected height simulation domain force direction respective numbers fluid cells cells direction practice calculation according lbgk equation split two steps first propagation step stream step propagates local pdfs cell along corresponding lattice link neighboring cell central one zero lattice velocity remains cell collide step relaxation towards local equilibrium performed cell set pdfs external force added yields pdfs succeeding time step feq step step derivation lattice boltzmann method see previous work free surface lbm main characteristic free surface model given dynamics second phase neglected assumes system two immiscible fluids dynamics first phase govern flow completely refer first phase liquid phase second phase gas phase layer two phases come figure free boundary interface cells real interface dashed line captured assigning interface cells liquid fraction contact free surface assumed thin relation spatial resolution separate regions liquid gas hand assumed large comparison assumption one simulate liquid phase modeling free surface special boundary condition means volume fluid approach introduces dynamic fill level lattice cell partial filling allowed cells free surface boundary fill level fraction cell volume currently filled liquid liquid mass cell may calculated cells liquid fraction thus form boundary fluid referred interface need special treatment outlined following refer cells interface cells fig shows interface layer liquid gas virtual free surface drawn dashed line stream step fill level may change due exchange pdfs neighboring cells mass balance two neighboring cells given gas liquid interface interface must fulfill characteristic free surface boundary gas flow neglected model pressure gas taken account interface two phases apart interface must freely follow liquid flow provided special free surface boundary condition assuming approximated normal vector free surface known interface cell position pdfs pointing towards liquid phase set feq feq stream step chosen matches pressure gas phase must given indeed show results zero strain rate tensor boundary seen substituting reconstructed pdfs formula momentum flux stress tensor second order moment pdfs feq feq feq feq feq last sum resembles system equilibrium follows normal vector obtained approximation gradient fill levels within local neighborhood cell also possible include effect surface tension directly boundary condition constant surface tension parameter given local curvature interface known gas pressure augmented laplace pressure pohl proposed method include surface tension approach extract local curvature surface normals local neighborhood second step normal computation order allow free advection free boundary gas liquid state interface cells must tracked carefully since interface cells represent boundary lbm scheme necessary assure closed layer interface cells around liquid cells facilitated set cell conversion rules thus direct state transition gas liquid allowed interface cell reaches fill level converted gas cell liquid cell respectively since conversion could introduce holes interface cell conversions either gas liquid interface cells may triggered newly generated interface cells initialized fill level respectively cell conversion scheme extended incorporation floating objects shown sec previous work particulate flow lbm walls solid obstacles block fluid motion must handled appropriate boundary conditions paper assume solid boundaries modeled condition relative motion liquid boundary zero achieved reflecting distribution functions hitting obstacle cell stream step opposed direction given obstacle cell bounced back according figure rigid body mapped lattice free surface flow four different cell types distinguished liquid interface gas obstacle velocity obstacle second term right hand side accelerates fluid according movement wall solid moving term vanish boundary handling reduced pure reflection boundary condition one directly obtain stresses exerted boundary known momentum exchange literature momentum transferred wall must equal change momentum results reflection pdf since elastic collisions assumed thus reflected change momentum simulation immersed rigid bodies behaving according newtonian physics facilitated discretizing shape body lattice every time step boundary set according object position orientation center cell point within volume represented shape body treated obstacle cell fig movement rigid bodies resulting stresses including resolution collisions realized practice coupling lattice boltzmann algorithm rigid body physics engine resulting body motion fed back fluid simulation setting boundary condition according cells covered obstacle hereby given velocity body given particle must behave according stresses exerted fluid net force torque particle obtained momentum flux according summed discretized particle surface let set grid points next discretized particle surface cell let set indices defined due reflection pdfs stated discrete surface integrals force torque respectively given obstacle liquid interface gas free surface conversions figure allowed state transitions cell types moving obstacles number transitions increases four ten free surface type conversions framed dotted line dashed arrows indicate critical state changes solid fluid latter sum center gravity object torque acting freely moving rigid body induce rotational movement body momentum exchange method become kind simulation particle suspensions lattice boltzmann mentioned large number publications slight modifications main difference works ladd aidun case obstacle cells always fluid blocking nodes mentioned references virtual fluid nodes inside discretized object shapes reactivated uncovered object approach however would difficult maintain presence flow since unclear interface would treated intersecting volume particle lbm free surface model sec additional presence moving obstacles sec two different boundaries liquid phase must treated dynamically fig illustrates situation box floating free surface liquid simulated according sec since four different types cell states consider liquid interface gas solid may change according fluid flow object movement need sophisticated cell conversion algorithm arises following show conversions organized way consistent boundary around liquid phase assured every time step fig shows total number conversions consists ten state transitions transitions bottom advection scheme free surface model sec need examination remaining transitions arise mapping solid grid assume velocity object small enough conversions obstacle occur positions neighborhood obstacle cells conversions obstacle one three fluid states liquid interface gas occur positions neighborhood fluid cells additional limitation direct consequence low mach number limitation lattice boltzmann scheme consider object penetrating free surface shown fig conversions obstacle cells generally safe conversions obstacle fluid could lead invalid lattice configurations holes interface layer thus correct cell type liquid interface gas needs determined valid lattice configuration adhered next time step achieved update rule based local neighborhood given obstacle cell define neighborhood cell obstacle prior obstacle cell converted fluid state determined follows contains liquid converted gas contains gas converted liquid interpolated set feq velocity point object surface closest cell center contains liquid gas converted interface interpolated cells set preceding case additionally fill level chosen interpolation fill levels interface cells also possible include gas cells pressure interpolation since pressure seen continuous quantity across free surface boundary free surface model sec gas phase taken account terms pressure force exerted onto free surface pressure forces however also acting surface solid bodies analogy boundary condition free surface use equilibrium function construct pdfs feq lattice links gas interface cells obstacle cell object momentum transferred body calculated value constructed pdf linearly interpolated liquid distribution function thus presence gas neighborhood obstacle cell momentum acting body locally given feq gas cell equilibrium function remains parts vanish simulation floating bodies following sections present results calculations performed method sec various simulations rigid body exposed free surface flow since many details method owed previous particulate flow simulations briefly recapitulate relevant results sec focus evaluation forces motion partially immersed rigid bodies sec order prove basic consistency method dealing floating objects free surface flows buoyancy forces discretized objects turned one main sources error last part sec apply method problem floating stability rigid bodies force computations spherical particles staircase approximation emphasized due discretization body shapes lattice sites described previous section certain priori error introduced issue particulate flows lot research regarding treatment complex curved boundaries lattice boltzmann last decades schemes allow second order spatial accuracy treatment curved boundaries directly applicable moving boundaries particles however kind flows going discuss final part paper interpolation based schemes like applied success particle simulations past turned little useful solid particle model described sec use throughout paper identical one used drag force computations suspended particle agglomerates net force deviation studied spherical particles stokes flow calculations performed various degrees spatial resolution set different lattice relaxation parameters sphere fixed center channel flow results found good agreement theoretical expectations second order boundary conditions surpassed staircase approximation tested circumstances sinking sphere simulations conducted order evaluate method moving particles proposed simulations included influence gravity used unified boundary treatment solid boundaries particle density accelerates frictional force imposed fluid surrounding balances weight terminal sinking velocity reached results showed small periodic oscillations measured force sphere moves velocity lattice error attributed staircase approximation boundary body fig shows volume error sphere radius traveling along path number cells actually covered sphere depending exact position relative lattice discretized shape particle changing set lattice links included discrete surface integral around boundary changes net force torque respectively interpolation based schemes applied however approximate curved boundary calculating exact distance cell center along lattice link reflection information used correct reflected pdfs hence error arising buoyancy force changes discrete volume object corrected purely local refinements suspensions without free surface problem buoyancy force fluctuations obstacles usually negligible corrected workarounds reported following subsections discuss issue free surface case forces buoyant rigid bodies due staircase approximation rigid body shapes buoyancy force depending object position partially immersed objects staircase function respect draft best explained cube side length volume per timestep volume discretized sphere time figure discretized volume spherical particle traveling along path velocity ideal volume graph repeats periodically every time steps liquid rest constant water level buoyant force given linear function respect draft discretization error exact lattice resolution keep buoyant objects coming rest equilibrium position especially objects planar faces cube would sometimes cause unphysical wobbling behavior object jumping two points slightly ideal floating position contrary fixed cuboid considered gauge liquid varied immersing less object surface changes buoyant force object would continuous volume interface treatment limited spatial accuracy lattice fill level taken account force calculations interface cells triple line liquid gas solid test cases linearly rising water level around partially immersed fixed cuboid led linear increase buoyant force free advection test test consistency unified algorithm free advection test performed channel flow constant homogeneous velocity laminar regime along gravitational field along spherical particle density placed freely floating center channel accelerated hydrodynamic forces acting surface fig gives impression simulated setup gas phase neglected model must hinder particle motion way means cases final velocity particle must equal fluid velocity systematic drag force evaluation cited preceeding section test might seem trivial first spot presence density gradient changes discretized particle shape sec least two important sources error consider hence following also examine results critical flow velocities order magnitude expected errors channel size measured lattice units particle diameter long sides channel moving walls order obtain homogeneous velocity profile across domain constant flow velocity along figure schematic view free advection test floating sphere laminar free surface flow channel driven moving wall boundary condition bottom longitudinal side planes channel entry outflow connected means periodic boundary condition left right boundaries connected periodically copying pdfs streamed outside one end cells opposing end domain whole process testing performed four different values namely gravitational constant four different channel velocities order check possible sources errors scenario run sequence increasing complexity starting simplified version simulation experiment first series following channel fully filled liquid without gas without gravitational field repetition without gas gravitational field stated particle density set case first run particle velocity matched liquid velocity last digit without deviations second run expecting errors due oscillations buoyant force described clearly measurable largest lowest lattice viscosity object velocity oscillated within interval larger values effect would damped factor case significant deviations velocity components particle next test series consisted channel partially filled liquid planar free surface influence gravity particle removed check behavior interface cells free surface remained planar tested circumstances expected lowest value locally spurious interface velocities occuring however absolute deviation bulk viscosity always restricted locally parts interfacial layer therewith critical low velocities finally tested advection solid body free surface flow placing particle density inside channel fluctuations lifting force comparable measured domain gravity tested cases however error also became visible components net force object thus introduced error streaming velocity particle believe caused errors introduced refilling step algorithm conversions occur object sticking interface layer described sec initialize cells local equilibrium parameters interpolated local neighborhood likely introduce certain error balanced asymmetrical case obstacle boundary gas liquid even case flows without gas phase problem initializing new liquid cells evaluated cases empty channel showed spurious currents velocities remaining cases sphere would follow current velocities given tab higher flow velocities lower values error would longer critical even apparent sphere velocities table errors velocity sphere lattice gravity critical low characteristic flow velocity order systematic error staircase approximated objects negligible long characteristic velocity flow large comparison however considered froude number scaling characteristic length simulations hand question arises whether accurate treatment particle boundaries achieved suffer presence hydrostatic density gradient unable find literature reports problem explicitly besides increasing spatial resolution applying local grid refinement methods entail complex restructuring whole implementation method found least one approach could provide significant improvement based originally chen volumetric interpretation lattice boltzmann method allows direct calculation momentum flux onto finite surface elements exploited realize boundary conditions like offer scale accuracy presented variant allows exact description surface obstacles including surface stress calculation independent surface motion free surface boundary treatment principle also volumetric approach would yield unification method described paper expect explore issue future publications stability floating bodies apply method problem floating stability rigid bodies archimedean law appears quite intuitive states buoyancy experienced immersed object equal weight liquid displaced immersed volume part body part water plane according rule body specific material density sink rise buoyant force gravitational force balance vertically practice however even balance forces given resulting equilibrium often found unstable consider instance cube shown fig positions exactly figure positions cube specific density hydrostatic equilibrium given however equilibrium left hand side unstable stable equilibrium found cube rotated degrees figure stable floating positions cube material density left right respectively time one corner cube place water line another one exactly figure unstable equilibrium cube slight horizontal displacement center buoyancy relative center gravity cube moment occurs pushes cube upright position center buoyancy constructed center gravity trapezoid immersed part cube makes together fluid surface construct base line segment elongated one direction parallel opposite segment latter elongated opposite direction base line next shifted end points line segment connected intersection connecting line line midpoints gives center gravity trapezoid figure box rotating longitudinal axis rolling motion half body volume immersed thus body floats hydrostatic equilibrium however shown right hand side heel angle degrees stable see consider cube slightly rotated upright position central axis dashed line fig heeled small angle center gravity cube point gravitational force acts vertically downward center buoyancy defined center gravity immersed part body slightly displaced left shown fig thus acting vertically upward rotational moment heeling direction occurs pushing cube upright position way shown starting degree rotated position right fig small angles heel position lead righting moment opposite direction result stable equilibrium position fig shows two stable floating positions cube densities equilibrium position floating cubes following check rotational equilibrium floating cubes density simulations given densities stable angle heel found upright axis cube shown fig cube density calibrate since dealing objects cube side length represented cuboid dimensions rotate around longitudinal axis schematic fig inside domain cells lower half filled liquid lattice viscosity initialized hydrostatic pressure three boxes placed inside basin slightly heeled upright position angle simulation run fluid motions ceased objects came rest fig shows visualizations simulated scenario various time steps since initial positions objects far equilibrium state motions would cause wavy disturbance liquid basin clearly seen visualizations even low resolution strong convergence towards ideal equilibrium given righting stability moment structures consider floating structure heeled certain angle upright axis intersection line direction upright axis body defined metacenter lies center gravity floating position body stable otherwise unstable see fig concept originates marine engineering commonly used characterization floating stability ships offshore structures comprehensive discussion found figure box objects density respectively striving towards vertical rotational equilibrium first three pictures show system motion last picture taken visible motions ceased snapshots visualized simulation output ray tracer lever arm torque arising horizontal displacement mass centers given sin hence righting stability moment body calculated sin curve sin called stability curve floating object curve derived analytically many relevant situations depending shape immersed structure important case structure points structure surface sinking rising water plane upon considered angles heel parallel opposing sides structure upright structure given according scribanti formula states center buoyancy upright position moment inertia water plane stands immersed volume whole body case cuboid length width height draft obtained position formula simplified putting keel point cuboid expression introduced ideal stability curve arbitrary cuboid calculated fig shows schematic stability curves cuboid structures density various width height ratio case cube stability curve negative slope origin shows upright position unstable equilibrium increased width yields higher floating stability test force calculation floating bodies righting stability moment floating cube different angles heel measured lattice boltzmann simulations compared ideal stability curve structure following cases gravitational constant chosen partially filled basin initialized pressure gradient according hydrostatic equilibrium relaxation time initial width height ratio box second pass lower ratio chosen time error examined three different resolutions box size lattice units respectively length side along axis rotation always chosen box positioned lattice exactly relative free surface plane simulation box fixed constant position angle heel box varied around longitudinal axis box center gravity figure stability curves various floating cuboids density negative slope graph cube case width height ratio angle heel deduced upright position unstable stability increases width enlarged figure stability curve heeled cuboid different resolutions legend reads width box lattice units different resolutions significantly less deviations torque stable box width per height ratio compared box shape ratio width height width heigh ideal ideal torque torque heeling angle time fig shows simulation results compared ideal stability curve error much larger second case since stability lever arm box shorter lower width per height ratio thus even small errors box discretization visible influence behavior box however cases seen errors decreasing higher resolutions convergence towards ideal values clearly given conclusion paper described method simulation flows means lattice boltzmann method based previous works momentum exchange method used obtain fluid stresses surface rigid bodies allows calculate resulting net force torque novel set cell conversion rules sec allows unrestricted movement solid bodies within domain including penetrations free surface demonstrated consistency method free advection test particle following homogeneous free surface channel flow influence gravity sec validation force torque calculations rigid bodies applied method classic mechanical problem floating stability immersed structures basic convergence towards equlibrium state successfully checked case box objects square sides took advantage fundamental stability formulae structures validated righting moment box shaped objects varying angles heel results found converge towards ideal values increased spatial resolution next step would interesting move hydrostatic stability examples towards dynamic ones include coupled interaction liquid solid objects present results together method improvements future publications acknowledgements simulations presented paper walberla lattice boltzmann framework rigid body phisics engine used software projects collaborative effort chair system simulation university author would like thank jan klaus iglberger daniela anderl stefan donath matthias markl tobias preclik florian schornbaum various discussions corrections finally many thanks christian kind correspondence via electronic mail references aidun ding direct analysis particulate suspensions inertia using discrete boltzmann equation fluid mech cyrus aidun jonathan clausen method complex flows annu rev fluid cyrus aidun yannan lattice boltzmann simulation solid particles suspended fluid journal statistical physics cyrus aidun yannan extension lbm direct simulation suspended particles near contact journal statistical physics bogner simulation floating objects flows master thesis university lehrstuhl informatik systemsimulation institut informatik bouzidi firdaouss lallemand momentum transfer fluid boundaries physics fluids buick greated gravity lattice boltzmann model phys rev caiazzo analysis lattice boltzmann nodes initialization moving boundary problems progress computational fluid dynamics hudong chen volumetric formulation lattice boltzmann method fluid dynamics basic concept physical review hudong chen chris teixeira kim molvig realization fluid boundary conditions via discrete boltzmann dynamics international journal modern physics binder christian feichtinger christian schmid nils peukert wolfgang ulrich simulation hydrodynamic drag aggregated particles journal colloid interface science captain derrett barrass ship stability masters mates edition fekken veldman buchner simulation green water loading using equations piquet editor proceedings international conference numerical ship hydrodynamics geert fekken numerical simulation flow moving rigid bodies phd thesis rijksuniversiteit groningen jan goetz massively parallel direct numerical simulation particulate flows phd thesis university dieter molekulare gasdynamik springer german luo theory lattice boltzmann method boltzmann equation lattice boltzmann equation phys rev christian kinetic approaches simulation free surface flow problems civil environmental engineering phd thesis technische braunschweig massie offshore hydromechanics delft university technology http iglberger klaus nils ulrich simulation moving particles lattice boltzmann method computers mathematics applications thies hofmann lattice boltzmann model free surface flow modeling foaming journal statistical physics ladd numerical simulations particulate suspensions via discretized boltzmann equation part theoretical foundation lawrence livermore national laboratory ladd numerical simulations particulate suspensions via discretized boltzmann equation part numerical results lawrence livermore national laboratory university lehrstuhl matik rigid body physics engine http inforurl university lehrstuhl informatik walberla lattice boltzmann framework url http luo unified theory lattice boltzmann models nonideal gases phys rev luo theory lattice boltzmann method lattice boltzmann models nonideal gases phys rev mei shyy luo force evaluation lattice boltzmann method involving curved geometry physical review persistence vision pty http persistence vision raytracer pohl high performance simulation free surface flows using lattice boltzmann method phd thesis university mei shyy luo lattice boltzmann method flows curved boundary journal computational physics rhode derksen van den akker volumetric method calculating flow around moving objects schemes physical review sauro succi lattice boltzmann equation fluid dynamics beyond oxford science publications verberg ladd model boundary conditions physical review letters mei shyy unified boundary treatment lattice boltzmann aerospace sciences meeting exhibit american institute aeronautics astronautics
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jun domain specific language modular knitting pattern definitions purl chelsea battell mcmaster university april contents introduction knitting pattern example verification example purl compiler overview error handling elements knitting ast node verification html generation ast node verification html generation row ast node verification html generation row elements ast node verification basic stitches ast node verification html generation compound stitch ast node verification html generation fixed stitch repeat ast node verification html generation undetermined stitch repeat ast node verification html generation ast node verification html generation join ast node verification html generation pattern ast node verification html generation pattern body ast node html generation row repeats ast node html generation pattern section ast node verification html generation pattern sample sample definition sample call root ast node html generation lexical analysis parser pass ast construction pass sample substitution pass pattern verification code generation html discussion appendices extra productions expressions condition compiler types parser types symbol lookup types examples tests test page dom style sheets test page styles target language styles purl language used modular definition verification knitting patterns syntax similar standard knitting pattern notation provided craft yarn council see purl provides constructs available standard notation allow reuse segments patterns report written using literate programming see tool literate programming eclipse lep see compiler source code html css web page rudimentary ide purl tangled lep plugin eclipse ide introduction knitting process strand yarn turned flexible fabric patterns written record knitting designs techniques simple project patterns using different stitches may shared orally knit complicated objects becomes necessary document instructions earliest known example object knit using stitches knit stitch dates century purl stitch yarn see popular standard instruction reference knitting written lists stitches stitch reference published last five years lists stitches basic included project improvements pattern documentation techniques tools necessary conducive increased complexity innovation knitting patterns patterns typically written according standard may include extra information number active stitches needles end row extra information guides knitter help avoid mistakes assumes pattern errors common case including industrial settings every pattern written pattern designer experiment perform many tedious computations make sure pattern correct also even though segments pattern may reused many others rewritten new pattern project implements compiler language called purl syntax similar standard knitting pattern notation purl features increase reusability segments knitting patterns compiler performs automatic verification number stitches row displays guiding information output pattern knitting pattern example knitting pattern bag standard notation pattern broken two sections body handle section begins add initial stitches needle body stitches handles rows pattern defined next section ends either join single asterisk means repeat stitches directed two asterisks means repeat rows betwen double asterisk row repeat instructions market bag body cast rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rnd rep end rep end rep end rep end rep end rep end rep end rep end rep end rep end rep end rep end rep end rnd rep end rnd rep end rep rnd rep end rnd rep end rep handle pick body top row rep end row rep end rep row row rep end row rep end rep row row rep end row rep end rep body top pattern example market bag program purl generate pattern blocks beginning sample new construct introduced increase reusability parts knitting patterns samples could used many patterns definition market bag pattern samples becomes much simpler sample max max rnd rnd end max sample diag onalla rnd end rnd end repeat sample type type row end row end repeat type rnd end rnd end repeat market bag body rnd end rnd end dia gonalla handle body top row row body top purl example market bag pattern constructs used example discussed detail throughout report verification example section body bag number stitches increasing last couple rows difficult knitter keep track number stitches needle rows pattern standard notation could written stitch count end row row rep end pattern example stitch count helpful discussed intro tedious necessary pattern designer figure value every row purl compiler computes value ensures row uses stitches row displays end every row reference knitter purl compiler overview purl compiler consists parser generates knitting pattern standard notation html first pass constructs abstract syntax tree according provided grammar implemented using recursive descent parsing techniques described lexer module used pass errors warnings reported first pass lexical syntactic errors pass two traverses syntax tree depth first replacing variables constructs purely elements purl expressible standard notation constructs discussed exploring individual pattern elements third pass syntax tree traversed depth first verification occurs pass errors indicate problems structure knitting pattern global state object used throughout parsing track information necessary error reporting section name position code row number generated pattern also used verification pass track pattern orientation width row index update nodes values necessary reason breaking parsing three passes syntax tree representation pattern much easier manipulate verify main feature purl ability define modular parametrized segments patterns pattern sample construct introduced language see second pass used trimming nodes representing sample calls also challenges verifying pattern necessary every row works stitches previous row pattern constructs work number stitches depends width current row since allow modular pattern definitions parameterized segments patterns verification done single pass source language error handling explanations many strategies used compiler handling errors including panicmode recovery approximately discussed covered avoid redundant explanations later whenever character symbol expected characters considered likely errors likely errors generate warning compilation continue usual expected likely typo keyword expected ident found typo assumed case warning created compilation continues unexpected symbols generate error lexer scan end production usually period comma symbols return node resume compilation next sibling certain stitches noted later reliable symbol delimiting siblings case lexer scan end parent production unexpected keyword found error message reports invalid use keyword unexpected ident symbol found error message reports invalid use ident unexpected character symbol found error message reports invalid use character unexpected symbol error sym type symtype addmsg msgtype error node invalid use ident sym value hasownvalue keywordsym sym type addmsg msgtype error node invalid use keyword sym value hasownvalue charsym sym type addmsg msgtype error node invalid use sym value character sed ast construction pass page pattern parse pattern page pattern parse title page colon separator page parse keyword page parse value page period terminator page parse keyword page parse value page parse origin page parse origin page row definition parse row type page row element parse page stitch parse page undetermined stitch repeat parse open page undetermined stitch repeat parse close page undetermined stitch repeat parse page undetermined stitch repeat parse repeat instruction page fixed stitch repeat parse open page fixed stitch repeat parse close page compound stitch parse open page compound stitch parse close page parse page parse count page join parse keyword page join parse count page join parse destination page join parse destination page page page section parse section page section parse title page section content parse page section content parse page sample def parse sample page sample def parse params page sample def parse params page sample call parse ident page sample call parse params page elements knitting section explores element knitting pattern corresponding representation throughout compilation organization allows additions new features single location provides isolated explanations knitting concept begin knitting project necessary add stitches needles called essentially number loops made one needles loop connected adjacent loops ast node hcoi hnati circular provisional syntax similar standard notation replacing also commonly used patterns requiring sts explicit purl example market bag body parse var coparse var node type nodetype caston value cast parse keyword cast parse value cast parse cotype period terminator node sed ast construction pass page ncluded locks page page page page keyword used declare pattern parse keyword sym type keywordsym caston nextsym sym type symtype var msg cast dedlaration must start keywordsym caston addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym caston start cast declaration addmsg msgtype error node msg scantosym charsym period nextsym node sed parse page ncluded locks page following keyword natural number given number stitches added needle parse value sym type symtype nat node value sym value nextsym else unexpected symbol addmsg msgtype error node missing cast count scantosym charsym period nextsym node sed parse page ncluded locks page default type flat grammar also allows circular provisional circular used knitting object beginning knitting center circle provisional used removed later project worked opposite direction parse cotype sym type keywordsym castoncirc node cotype cotype nextsym sym type keywordsym castonprov node cotype cotype nextsym else node cotype cotype sed parse page end marked period period terminator sym type charsym period nextsym sym type charsym addmsg msgtype warning node use symbol end node type nextsym else unexpected symbol addmsg msgtype error node missing symbol end node type scantosym charsym period nextsym sed parse page parse page row definition parse page parse page join parse page sample call parse page ncluded locks page verification verified initial side row width row index set state object verify var verifycaston node sidetype width node value rowindex sed verification pass page html generation write html var writeco node var cotype node cotype node cotype node cotype addelement tagtype div classtype caston cast node value sts cotype sed page possible begin knitting completed knitted object picking stitches done using one needle pull loops yarn spaces along edge stitches picked new active stitches needle ast node hpui hnati hstringi body top purl example market bag handle parse var puparse var node type nodetype pickup value pick parse keyword pick parse value pick parse origin period terminator node sed ast construction pass page ncluded locks page page page page keyword used beginning declaration stitches parse keyword sym type keywordsym pickup nextsym sym type symtype var msg pick dedlaration must start keywordsym pickup addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym pickup start pick declaration addmsg msgtype error node msg scantosym charsym period nextsym node sed parse page ncluded locks page following keyword natural number given number stitches added needle parse value sym type symtype nat node value sym value nextsym else unexpected symbol addmsg msgtype error node missing pick count scantosym charsym period nextsym node sed parse page ncluded locks page next expect keyword followed string directing stitches picked parse origin sym type keywordsym nextsym else unexpected symbol scantosym charsym period nextsym node sym type symtype node sym value nextsym else unexpected symbol addmsg msgtype error node missing pick origin scantosym charsym period nextsym node sed parse page ncluded locks page page end marked period verification verified initial side row width row index set state object verify var verifypickup node sidetype width node value rowindex sed verification pass page html generation write html var writepu node addelement tagtype div classtype caston pick node value sts node sed page row stitches added needles possible begin body pattern typically knitting worked left needle right needle knitting hand may done flat round methods worked horizontally meaning object created sequence knit rows flat knitting done two straight needles thought two stacks since first stitch worked row last created previous row circular knitting circular needle two needles connected cable thought two queues since first stitch worked row first stitch created previous row active stitches loops needles left needle start row considered active removed needle could pulled whatever stitches support stitch worked pulling loop yarn active stitch dropping active stitch left needle previously active stitch anchored loop pulled loop active stitch right needle row list stitches required work active stitches left needle right needle ast node hrowdef row rnd hrowelemi hrowelemi row definition parse var rowdefparse var node type nodetype row row parse row type row parse color colon row parse row elements period terminator node sed ast construction pass page ncluded locks page page page page page row begins row type either row rnd flat knitting row type row used meaning stitches left needle worked popped whole object turned left right needles swapped process begin last stitch pushed right needle first popped left circular knitting row type rnd used meaning stitches left needle worked new stitches right needle stitches moved along cable worked fifo fashion default row type row assumed unexpected symbol found body section market bag pattern example knit round handle knit flat rnd end purl example market bag body row row purl example market bag handle row row definition parse row type sym type keywordsym row node rowtype rowtype row nextsym sym type keywordsym rnd node rowtype rowtype rnd nextsym else unexpected symbol addmsg msgtype error node invalid row type specified node rowtype rowtype rnd nextsym sed row definition parse page ncluded locks page pattern writer optionally specify yarn color used row currently main color contrasting color available language main color assumed default row definition parse color sym type keywordsym colormain node colortype main nextsym sym type keywordsym node colortype nextsym sed row definition parse page colon symbol separates row declaration content colon separator sym type charsym colon nextsym sym type charsym semicolon var msg use symbol listing node type elements addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing symbol listing node type elements addmsg msgtype error node msg scantosym charsym period sed pattern parse page row definition parse page section parse page sample branches parse page sample def parse page ncluded locks page elements row pattern separated commas syntax tree node row element added array representing children row row definition parse row elements node push rowelemparse sym type charsym nextsym node push rowelemparse sed row definition parse page end row definition marked period verification verify row first node needs row index side properties set used pattern output verify row setup node index rowindex node sed verify row page row verification function uses rowstate object updated upon verification child row stitches compound stitches stitch repeats verify row children var rowstate width workedst stchange verifyrowelemchildren node rowstate sed verify row page details verification specific row elements discussed row elements section see verify row elem children node var verifyrowelemchildren node var workedst stchange node var node verifyrowelem node workedst workedst stchange stchange sed verification pass page verification error caught level incorrect number worked stitches means stitches specified row either use less require stitches exist previous row verify row worked stitches error rowstate workedst width var msg rowstate workedst sts worked width sts addmsg msgtype node msg sed verify row page children row verified row width node state object updated according stitch change caused row row type row row project flipped end row side changed also incremented one move next sibling verify row state update node width rowstate workedst rowstate stchange width node width node rowtype rowtype row sidetype sidetype sidetype sidetype rowindex sed verify row page verify row var verifyrow node row setup row children row worked row update sed verification pass page ncluded locks page page page page html generation written row includes row type side color explicitly given row index children number active stitches end row write html row var writerow node var push openelement tagtype div classtype row node rowtype rowtype row push row node rowtype rowtype rnd push rnd push node index node colortype main push node colortype push push node var node var node push writenode node push var count node width sts push addelement tagtype span classtype count push closeelement tagtype div return sed page row elements children row grouped row elements rowelem production used provide structure language necessary allow nesting undetermined stitch repeats see desirable allow fixed repeats see fixed stitch repeats children row elements basic stitch compound stitch fixed stitch repeat undetermined stitch repeat ast node hrowelemi hstitchopi hustrepi hstitchopi hfixedstrepi hcompsti hbasicsti parse function corresponding rowelem production determine rowelem current symbol first symbol passes result appropriate parse function current symbol basic stitch open angle bracket open parentheses need parse according stitchop production asterisk possible first symbol undetermined stitch repeat asterisk parse according ustrep production row element parse var rowelemparse var node hasownvalue stitchsym sym type sym type charsym openangle sym type charsym openbrack node sym type charsym node ustrepparse else unexpected symbol addmsg msgtype error node invalid row element scantosym charsym period node sed ast construction pass page ncluded locks page stitch parse var var node sym type charsym openbrack node fixedstrepparse sym type charsym openangle node compstparse hasownvalue stitchsym sym type node else unexpected symbol var msg sym value known stitch start stitch repeat compound stitch addmsg msgtype error node msg scantosym charsym period node sed ast construction pass page ncluded locks page verification row elements verified according row element node type details covered respective sections verify row elem var verifyrowelem node var workedst stchange var rep node repcount node repcount node repcount value rep node repcount value var num node num node num node num value num node num value switch node type nodetype fixedstrep fixed repeat break nodetype ustrep undetermined repeat break nodetype compst compound break nodetype knit nodetype nodetype knittbl nodetype purltbl nodetype knitbelow nodetype purlbelow nodetype nodetype slipkw nodetype slippw nodetype yarnover nodetype knitfb nodetype purlfb nodetype make nodetype makel nodetype maker nodetype knittog nodetype purltog nodetype ssk nodetype ssp nodetype psso break default break sed verification pass page ncluded locks page page page page basic stitches discussed stitch typically formed pulling trailing yarn active stitch left needle creating new stitch right needle active stitches left needle right needle passes dropped left needle stitch complete variations direction right needle passes last active stitch whether yarn front behind needle allow different stitches created complex stitches created pulling trailing yarn location fabric multiple active stitches rather single active stitch stitch adds new stitches right needle removed left called increase similarly stitch adds fewer stitches right needle drops left called decrease say number stitches dropped left needle number worked stitches since number stitches previous row processed say difference stitch makes width stitch change due desire adhere standard knitting pattern stitch notation allow flexibility naming identifiers language lexing parsing stitches treated differently rest language stitches numeric parameters middle stitch notation consider stitch represents knit together stitch natural number valid string form allowed structure identifiers acceptable identifier reserve certain strings parameterized segments stitches attempt made match identifier string regular expressions stitch assuming ident symbol stitch type determined lexer specific information contained stitch string extracted stitch parse function example row market bag pattern colon list stitches created row row purl example market bag handle row ast node stitch information var sym value sed basic stitch parse page stitch may optionally followed natural number expression indicate number times stitch repeated node rep count optional sym type symtype sym type symtype nat node repcount sed compound stitch parse page basic stitch parse page basic stitch parse var var node switch sym type stitchsym knit knit break stitchsym break stitchsym knittbl knittbl break stitchsym purltbl purltbl break stitchsym knitbelow knitbelow break stitchsym purlbelow purlbelow break stitchsym break stitchsym slipkw slipkw break stitchsym slippw slippw break stitchsym yarnover yarnover break stitchsym knitfb knitfb break stitchsym purlfb purlfb break stitchsym make make break stitchsym makel makel break stitchsym maker maker break stitchsym knittog knittog break stitchsym purltog purltog break stitchsym ssk ssk break stitchsym ssp ssp break stitchsym psso psso break default node workedst node stchange break nextsym node rep count optional node sed ast construction pass page ncluded locks page page page page page page page page page page page page page page page page page page page page page page knit ubiquitous stitch needle entry top active stitch left yarn position back effect knit node type nodetype knit node workedst node stchange sed basic stitch parse page knit back loop needle entry top active stitch right yarn position back effect knittbl node type nodetype knittbl node workedst node stchange sed basic stitch parse page knit stitch parameter num needle entry num stitches top active stitch right yarn position back effect knitbelow node type nodetype knitbelow node num node workedst node stchange sed basic stitch parse page purl needle entry top active stitch right yarn position front effect purl node type nodetype node workedst node stchange sed basic stitch parse page purl back loop needle entry top active stitch left yarn position front effect purltbl node type nodetype purltbl node workedst node stchange sed basic stitch parse page purl stitch parameter num needle entry num top active stitch right yarn position front effect purlbelow node type nodetype purlbelow node num node workedst node stchange sed basic stitch parse page slip needle entry top active stitch left yarn position back notes loop pulled effect slip node type nodetype node workedst node stchange sed basic stitch parse page slip knitwise default slip stitch needle entry top active stitch left yarn position back notes loop pulled effect slipkw node type nodetype slipkw node workedst node stchange sed basic stitch parse page slip purlwise needle entry top active stitch right yarn position front notes loop pulled effect slippw node type nodetype slippw node workedst node stchange sed basic stitch parse page yarn increase yarn position back instruction wrap yarn counterclockwise arount right needle effect yarnover node type nodetype yarnover node workedst node stchange sed basic stitch parse page knit front back increase instruction active stitch knit knit back loop effect knitfb node type nodetype knitfb node workedst node stchange sed basic stitch parse page purl front back increase instruction active stitch purl purl back loop effect purlfb node type nodetype purlfb node workedst node stchange sed basic stitch parse page make increase parameter num setup back left needle picks vertical bar top left top right active stitches instruction knit stitch picked left needle repeat num times effect make node type nodetype make node num node workedst node stchange sed basic stitch parse page make left default make stitch increase parameter num setup back left needle picks vertical bar top left top right active stitches instruction knit stitch picked left needle repeat num times effect makel node type nodetype makel node num node workedst node stchange sed basic stitch parse page make right increase parameter num setup front left needle picks vertical bar top left top right active stitches instruction knit back loop stitch picked left needle repeat num times effect maker node type nodetype maker node num node workedst node stchange sed basic stitch parse page knit together decrease parameter num needle entry top num active stitches left yarn position back effect knittog node type nodetype knittog node num node workedst node num node stchange node num sed basic stitch parse page purl together decrease parameter num needle entry top num active stitches right yarn position front effect purltog node type nodetype purltog node num node workedst node num node stchange node num sed basic stitch parse page slip slip knit decrease instruction slip knitwise slip purlwise knit two slipped stitches together effect ssk node type nodetype ssk node workedst node stchange sed basic stitch parse page slip slip purl decrease instruction slip knitwise slip purlwise purl two slipped stitches together effect ssp node type nodetype ssp node workedst node stchange sed basic stitch parse page pass slip stitch decrease needle entry right needle second active stitch top left yarn position back instruction pass stitch top active stitch right needle needle notes commonly used slip knit hence passing slipped stitch effect psso node type nodetype psso node workedst node stchange sed basic stitch parse page verification role basic stitch verification pass update row state stitch change number worked stitches given stitch verify basic stitch workedst node workedst rep stchange node stchange rep sed verify row elem page html generation writing stitches follows standard notation write html basic stitch var node var var rep var num node repcount node repcount value rep node repcount value node num node num value num node num value node num num node num switch node type nodetype knit push addelement tagtype span classtype rep break nodetype push addelement tagtype span classtype rep break nodetype knittbl push addelement tagtype span classtype rep tbl break nodetype purltbl push addelement tagtype span classtype rep tbl break nodetype knitbelow push addelement tagtype span classtype num rep break nodetype purlbelow push addelement tagtype span classtype num rep break nodetype push addelement tagtype span classtype rep break nodetype slipkw push addelement tagtype span classtype rep break nodetype slippw push addelement tagtype span classtype rep break nodetype yarnover push addelement tagtype span classtype rep break nodetype knitfb push addelement tagtype span classtype kfb rep push rep break nodetype purlfb push addelement tagtype span classtype pfb rep push rep break nodetype make push addelement tagtype span classtype num rep push rep break nodetype makel push addelement tagtype span classtype num rep push rep break nodetype maker push addelement tagtype span classtype num rep push rep break nodetype knittog push addelement tagtype span classtype num tog rep push rep break nodetype purltog push addelement tagtype span classtype num tog rep push rep break nodetype ssk push addelement tagtype span classtype ssk rep push rep break nodetype ssp push addelement tagtype span classtype ssp rep push rep break nodetype psso push addelement tagtype span classtype psso rep push rep break default break return sed page compound stitch possible work number stitches single stitch means loop pulled active stitch stitch dropped left needle remains sequence stitches performed ast node hcompsti hbasicsti hbasicsti hexpri example compound stitch means perform stitches angle brackets one stitch purl example compound stitch sequence stitches compound stitch contained angle brackets use bracket symbols generate warning allow compilation continue compound separated siblings comma children also case error necessary lexer scan terminator parent node production continuing parsing compound stitch parse open sym type charsym openangle nextsym sym type charsym openparen sym type charsym openbrace sym type charsym openbrack addmsg msgtype warning node use symbol start compound stitch nextsym else unexpected symbol addmsg msgtype error node missing symbol start compound stitch scantosym charsym period node sed compound stitch parse page ncluded locks page compound stitch parse close sym type charsym closeangle nextsym sym type charsym closeparen sym type charsym sym type charsym closebrack addmsg msgtype warning node use symbol end compound stitch nextsym else unexpected symbol addmsg msgtype error node missing symbol end compound stitch scantosym charsym period node sed compound stitch parse page ncluded locks page compound stitch brackets comma separated list basic stitches compound stitch parse children node push sym type charsym nextsym node push sed compound stitch parse page compound stitch optionally ends natural number variable indicating many times compound stitch repeated note compound stitch works one stitch every repeat since stitches sequence worked single active stitch left needle compound stitch parse var compstparse var node type nodetype compst compound parse open compound parse children compound parse close node rep count optional node sed ast construction pass page ncluded locks page page page page verification verify compound stitch verifyrowelemchildren node workedst rep stchange stchange rep sed verify row elem page html generation write html compound stitch var writecompst node var var node var node push writenode node push node repcount node repcount value push node repcount value times push next return sed page fixed stitch repeat sequence stitches may repeated fixed number times ast node hfixedstrepi hstitchopi hstitchopi hexpri example means perform sequence brackets knit purl three times purl example fixed repeat children fixed stitch repeat sequence fixed stitch repeats compound stitches basic stitches contained within parentheses saw compound stitches confidently assume location next sibling error situation scan terminator parent node production fixed stitch repeat parse open sym type charsym openbrack nextsym sym type charsym openangle sym type charsym openbrace sym type charsym openparen addmsg msgtype warning node use symbol start fixed stitch repeat nextsym else unexpected symbol addmsg msgtype error node missing symbol start fixed stitch repeat scantosym charsym period node sed fixed stitch repeat parse page ncluded locks page fixed stitch repeat parse close sym type charsym closebrack nextsym sym type charsym closeangle sym type charsym sym type charsym closeparen var msg use symbol end fixed stitch repeat stitches addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing symbol end fixed stitch repeat stitches addmsg msgtype error node msg scantosym charsym period node sed fixed stitch repeat parse page ncluded locks page children fixed stitch repeat node commma separated fixed stitch repeat parse children node push sym type charsym nextsym node push sed fixed stitch repeat parse page fixed stitch repeat must end natural number expression fixed stitch repeat parse var fixedstrepparse var node type nodetype fixedstrep fixed repeat parse open fixed repeat parse children fixed repeat parse close node repcount charsym period node sed ast construction pass page ncluded locks page page page verification fixed stitch repeat adds number worked stitches stitch change multiplied specified repeat amount values parent node verify fixed stitch repeat verifyrowelemchildren node workedst workedst rep stchange stchange rep sed verify row elem page html generation write html fixed stitch repeat var writefixedstrep node var node var node push writenode node node repcount value times sed page undetermined stitch repeat sequence stitches may repeated number times depends current length row call undetermined stitch repeat ast node hustrepi hstitchopi hstitchopi last hexpri end first example means perform knit stitch many times required get end row second example means perform knit stitch last two stitches purl last two end purl example undetermined stitch repeat purl example undetermined stitch repeat undetermined stitch repeat must begin asterisk compound stitches fixed repeats invalid symbol used lexer scans terminator parent production continue parsing undetermined stitch repeat parse open sym type charsym nextsym else unexpected symbol var msg missing symbol start undetermined stitch repeat addmsg msgtype error node msg scantosym charsym period node sed undetermined stitch repeat parse page ncluded locks page following asterisk sequence basic stitches compound stitches fixed stitch repeats undetermined stitch repeat parse children node push sym type charsym nextsym node push sed undetermined stitch repeat parse page sequence children terminated semicolon undetermined stitch repeat parse close sym type charsym semicolon nextsym sym type charsym colon var msg use symbol end undetermined stitch repeat stitches addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing symbol end undetermined stitch repeat stitches addmsg msgtype error node msg scantosym charsym period node sed undetermined stitch repeat parse page ncluded locks page error remainder undeterminded stitch repeat since past sequence children assume next comma delimits next sibling next period terminator parent production next pattern writer must specify far along row repeat performed beginning keyword undetermined stitch repeat parse sym type keywordsym nextsym sym type symtype var msg use keywordsym undetermined stitch repeat addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym undetermined stitch repeat addmsg msgtype error node msg scantosym charsym charsym period sed undetermined stitch repeat parse page ncluded locks page keyword end used sequence repeated end row keyword last used sequence repeated within given number stitches end row reason nesting undetermined stitch repeats possible important note number stitches remaining left needle end last repeat must coincide exactly number specified end undetermined stitch repeat parse repeat instruction sym type keywordsym nextsym node num charsym charsym period sym type keywordsym end node num type nodetype value nextsym else unexpected symbol var msg missing repeat instructions expecting keywordsym keywordsym end addmsg msgtype error node msg scantosym charsym charsym period sed undetermined stitch repeat parse page ncluded locks page undetermined stitch repeat parse var ustrepparse var node type nodetype ustrep undetermined repeat parse open undetermined repeat parse children undetermined repeat parse close undetermined repeat parse undetermined repeat parse repeat node sed ast construction pass page ncluded locks page page page page page verification verify undetermined stitch repeat first determine number stitches children repeated number stitches worked single repeat divide value number unworked stitches equal division remainder end row case error generated otherwise row state parent node updated number worked stitches stitch change caused undetermined stitch repeat verify undetermined stitch repeat verifyrowelemchildren node var sttowork workedst node num value var remainst sttowork workedst remainst var msg remainst remain last possible repeat addmsg msgtype node msg else rep sttowork sttowork workedst workedst workedst workedst rep stchange stchange rep sed verify row elem page html generation write html undetermined stitch repeat var writeustrep node var var node var node push writenode node push rep var rem node num value rem push end rem push last rem rem push last rem sts else push invalid value return sed page considered rows stitches elements followed sufficient construct simple pattern ast node hboi hnati purl example market bag body keyword used declare pattern parse sym type keywordsym bindoff nextsym sym type symtype var msg bind declaration must start keywordsym bindoff addmsg msgtype warning node msg else unexpected symbol var msg missing keywordsym bindoff start bind declaration addmsg msgtype error node msg scantosym charsym period nextsym node sed parse page ncluded locks page next natural number given number stitches parse count sym type symtype nat node value sym value nextsym else unexpected symbol addmsg msgtype error node bind count specified scantosym charsym period nextsym node sed parse page ncluded locks page ends period terminator parse var boparse var node type nodetype bindoff value parse parse count period terminator node sed ast construction pass page ncluded locks page page page verification value node must equivalent final width row otherwise either stitches still active completing pattern else stitches former error much severe verify var node node value width var msg binding node value sts width sts addmsg msgtype node msg sed verification pass page html generation write html var writebo node var msg bind node value sts addelement tagtype div classtype bindoff msg sed page join alternative using finish pattern join remaining active stitches location section another knitted object active stitches joined active stitches called grafting ast node hjoini join hnati hstringi body top purl example market bag handle join join keyword used declare join pattern join parse keyword sym type keywordsym nextsym sym type symtype var msg join declaration must start keywordsym addmsg msgtype warning node msg else unexpected symbol var msg missing keywordsym start join declaration addmsg msgtype error node msg scantosym charsym period nextsym node sed join parse page ncluded locks page next natural number given number stitches join join parse count sym type symtype nat node value sym value nextsym else unexpected symbol addmsg msgtype error node join count specified scantosym charsym period nextsym node sed join parse page ncluded locks page necessary state active stitches joined begins keyword followed string directions join join parse destination sym type keywordsym nextsym else unexpected symbol scantosym charsym period nextsym node sym type symtype node sym value nextsym else unexpected symbol addmsg msgtype error node missing join destination scantosym charsym period nextsym node sed join parse page ncluded locks page page join ends period terminator join parse var var node type nodetype value arse keyword arse count arse period terminator node sed ast construction pass page ncluded locks page page page page verification value join node must equivalent final width row otherwise either stitches still active completing pattern else stitches former error much severe verify join var node node value width var msg joining node value sts width sts addmsg msgtype node msg sed verification pass page html generation write html join var node var msg join node value sts node addelement tagtype div classtype msg sed page pattern since seen enough construct simple patern look syntax knitting pattern written purl ast node hpatterni pattern hstringi hcoi hbodyi hboi one row row purl example simple single row pattern pattern definition begins keyword pattern pattern parse pattern sym type keywordsym nextsym sym type symtype var msg pattern declaration must start keywordsym addmsg msgtype warning node msg nextsym else unexpected symbol var msg expecting keywordsym start pattern declaration addmsg msgtype error node msg scantosym charsym colon sed pattern parse page ncluded locks page next requirement string double quotes title patern reason using string rather single identifier allow greater flexibility naming patterns way pattern name also include reserved keywords identifier provided rather string warning created compilation continues symbol causes error lexer scans next colon symbol required following pattern title pattern parse title sym type symtype node name sym value nextsym sym type symtype node name sym value var msg remember use double quotes around name pattern addmsg msgtype warning node msg nextsym else unexpected symbol addmsg msgtype error node pattern name specified scantosym charsym colon sed pattern parse page ncluded locks page main content pattern may pattern body pattern number defined sections symbol colon keyword parsing simple pattern simple pattern provides instructions single discrete object contrast composite pattern provides instructions knit multiple objects joined form larger structure pattern parse main sym type keywordsym caston content parse else parse composite sed pattern parse page ncluded locks page page simple pattern begins followed pattern body finally join pattern content parse node coparse node bodyparse node boparse sed pattern parse main page pattern parse var var node type nodetype parse parse colon parse main node sed ast construction pass page ncluded locks page page page page verification property null verifying simple pattern verify children otherwise composite pattern verify children pattern sections case verify pattern var node node verifynode node node verifynode node else node sed verification pass page html generation write html pattern var node var push openelement tagtype div classtype push addelement tagtype div classtype patternname node name node push writenode node push writebody node push writenode node node var node push writenode node push closeelement tagtype div sed page pattern body body pattern main content pattern everything consists zero rows row repeats see sample calls see note pattern production uses body production create child nodes row repeats sample definitions see constructs use body production ast node hbodyi hrowdef hrowrepi hsamplecalli body parse method builds returns array nodes make pattern body body parse var bodyparse var bodyelems sym type symtype eof switch sym type keywordsym row keywordsym rnd bodyelems push rowdefparse break symtype rowrep bodyelems push rowrepeatparse break symtype bodyelems push samplecallparse break default bodyelems bodyelems sed ast construction pass page html generation write html body var writebody node var push openelement tagtype div classtype body node var node push writenode node push closeelement tagtype div sed page row repeats elements pattern body repeated number times rather rewriting elements row repeat may used concise notation ast node hrowrepeati hbodyi repeat hexpri body diagonallace sample used market bag pattern uses row repeat means two rows double asterisk repeat repeated times rnd end rnd end repeat purl example ribbing repeat row repeat begins two asterisks sym type symtype rowrep nextsym sym type charsym var msg row repeat must begin symtype rowrep addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing symtype rowrep start row repeat addmsg msgtype error node msg scantosym charsym period nextsym sed row repeat parse page ncluded locks page end row repeat body marked repeat keyword must followed natural number variable sym type keywordsym repeat nextsym sym type symtype var msg row repeat body must followed keywordsym repeat addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym repeat row repeat body addmsg msgtype error node msg scantosym charsym period nextsym sed row repeat parse page ncluded locks page children row repeat represented array nodes use body parse method simple patterns pattern sections row repeat parse var rowrepeatparse var node type nodetype rowrep node bodyparse node repcount charsym period node sed ast construction pass page ncluded locks page page html generation write html row repeat var writerowrep node var push openelement tagtype div classtype rowrep push push writebody node push rep node repcount value times push closeelement tagtype div sed page pattern section writing pattern create multiple discrete objects pattern must made number pattern sections case write composite pattern pattern zero pattern sections children pattern parse composite sym type keywordsym node push sed pattern parse main page ast node hsectioni section hstringi hcoi hpui hbodyi hboi hjoini market bag body section market bag pattern body rnd end rnd end dia gonalla purl example market bag body section definition begins section keyword section parse section sym type keywordsym nextsym sym type symtype var msg section declaration must start keywordsym addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym start section declaration addmsg msgtype node msg scantosym charsym colon sed section parse page ncluded locks page title section follows string enclosed double quotes title pattern identifier symbol found instead warning generated name section set identifier compilation continue symbol cause error lexer scan colon separator occur section content section parse title sym type symtype node name sym value sectionname node name nextsym sym type symtype node name sym value var msg remember use double quotes around name section addmsg msgtype warning node msg nextsym else unexpected symbol addmsg msgtype error node section name specified scantosym charsym colon sed section parse page ncluded locks page parsing content section similar parsing content simple pattern except picking stitches joining alternatives respectively section content parse sym type keywordsym caston node coparse sym type keywordsym pickup node puparse else unexpected symbol node scantosym charsym period nextsym node bodyparse sym type keywordsym bindoff node boparse sym type keywordsym node else unexpected symbol node scantosym charsym period nextsym sed section parse page ncluded locks page page section parse var var node type nodetype parse parse colon content parse node sed ast construction pass page ncluded locks page page page page verification pattern section verification similar verification simple pattern first sectionname property state object set name current section used reporting errors verify section var node sectionname node name verifynode node node verifynode node sed verification pass page html generation write html section var node var push openelement tagtype div classtype push addelement tagtype div classtype sectionname node name push writenode node push writebody node push writenode node push closeelement tagtype div sed page pattern sample pattern sample may thought segment knitting pattern purl construct allows pattern samples defined natural number parameters pattern sample called pattern body analogous concept standard notation sample definition order use sample must first defined outside pattern definition hsampledef sample hidenti hidenti hidenti hsamplebranchi hbodyi hsamplebranchi hconditioni hbodyi circle sample used market bag pattern example sample branch two parameters max given rows added calling pattern max sample max max rnd rnd end max purl example circle sample market bag pattern sample definition begins sample keyword sample def parse sample sym type keywordsym sample nextsym sym type symtype var msg sample definition must start keywordsym sample addmsg msgtype warning node msg nextsym else unexpected symbol var msg missing keywordsym sample start sample definition addmsg msgtype error node msg scantosym symtype sed sample def parse page ncluded locks page next sample identifier since sample part target language identifier seen compilation reason consider aesthetic motivations sample naming pattern section naming current symbol identifier error created lexer scans next colon symbol continue compilation sample def parse ident sym type symtype node name sym value nextsym hasownvalue keywordsym sym type var msg sym value reserved keyword valid sample identifier addmsg msgtype error node msg scantosym keywordsym charsym colon else addmsg msgtype error node missing invalid sample identifier scantosym keywordsym charsym colon sed sample def parse page sample definition uses parameters parameter list prefixed keyword followed one identifiers separated commas added list parameter names used sample definition sample def parse params sym type keywordsym nextsym sym type symtype node paramnames push sym value nextsym else unexpected symbol sym type charsym nextsym sym type symtype node paramnames push sym value nextsym else unexpected symbol sed sample def parse page ncluded locks page page sample definition node added collection samples global state object parser sample may used definition sample definition written use specific branch sample body branch condition satisfied sample without branching expect colon separator parameters followed sample definition body parsed way simple patterns pattern sections row repeats sample def parse var sampledefparse var node type nodetype sampledef paramnames children line state line sample def parse sample sample def parse sample def parse params samples node name node sym type charsym sample branches parse else colon node bodyparse sed ast construction pass page ncluded locks page page page page page sample defined branches branch begins followed condition colon separator sample body sample branches parse sym type charsym nextsym var branch type nodetype branch branch colon branch bodyparse node push branch sed sample def parse page ncluded locks page sample call sample defined called pattern body simple pattern section row repeat sample definition sample example section also includes recursive sample call hsamplecalli hidenti hexpri hexpri purl example use stockinette stitch sample sample call begins identifier sample used sample call parse ident sym type symtype node name sym value nextsym else unexpected symbol addmsg msgtype error node missing sample call identifier scantosym charsym period sed sample call parse page ncluded locks page given identifier details corresponding sample definition acquired global state object sample definition parameter list prefixed keyword loop array parameter names sample definition construct parameter map object assigns passed expressions parameter names incorrect number parameters given error created lexer scans end sample call sample call parse params var sampledef samples node name sym type keywordsym nextsym var var sampledef paramnames sym type charsym nextsym else unexpected symbol node parammap sampledef paramnames sampledef paramnames var msg node name parameters required passed addmsg msgtype error node msg scantosym charsym period sed sample call parse page ncluded locks page sample call terminated period symbol sample call parse var samplecallparse var node type nodetype samplecall parammap sample parse sample parse params period terminator node sed ast construction pass page ncluded locks page page page root looked available knitting pattern constructs purl remains combine top level elements knitting pattern program sample definitions pattern definition market bag pattern given introduction example complete purl program ast node hprogrami hsampledef hpatterni present implementation sample definitions one pattern definition must combined one file sample definitions pattern definition ideal since purl meant allow modularity reusability segments knitting patterns sufficient initial implementation experimentation program parse var programparse var program type nodetype root null sym type keywordsym sample sampledefparse program program sed ast construction pass page html generation write html root var writeroot node writenode node sed page lexical analysis function createlexer returns object consists function get next symbol function get current line code input first split array createlexer private method next update current character position within array line position line var var null var symtype eof var pos var linenum var var pos pos pos linenum linepos else symtype eof return lexer getline function lexer getsym function ncluded locks page page getline simply returns object properties line number line position character position lexer getline function getline num linenum pos charpos pos sed page getsym determines current token value current character skipping whitespace characters lexer getsym function getsym symtype eof lex eof symtype lex lex num lex alphanum else lex char type symtype unknown value symtype unknown sed page ncluded locks page page page page page lexer reaches end input end file token returned character longer updated lex eof type symtype eof value symtype eof sed lexer getsym function page current character double quote symbol type title everything next double quote value lex string var symtype symtype eof symtype type symtype value sed lexer getsym function page current character number symbol type natural number symbol value concatenation characters character lex num var num num num type symtype nat value num sed lexer getsym function page current character letter concatenate characters next character string matches reserved keyword return symbol representing keyword string matches regular expression stitch lookup return symbol representing stitch otherwise return ident symbol lex alphanum var var idsym keywordsym keywordsym idsym type value var stsym stitchsym stitchsym stsym type stitchsym stsym value type symtype ident value sed lexer getsym function page current symbol char symbol lookup asterisk first check next character also asterisk double asterisk row repeat symbol otherwise return initial character symbol type value lex char var chsym charsym charsym chsym var charsym charsym symtype rowrep charsym openangle charsym equal symtype lesseq charsym closeangle charsym equal symtype greatereq type value sed lexer getsym function page parser var ast pass sample pass pass var addmsg msgtype node msgstr var msgobj messagetype msgtype sectionname sectionname line state line rowindex rowindex message msgstr messages push msgobj switch msgtype msgtype node haserrormsg break msgtype warning node haswarningmsg break msgtype node break default break var return input sectionname null samples messages var input console pass json undefined state json undefined samplesubstitutionpass ast pass json undefined state json undefined verificationpass ast pass json undefined state json undefined messages messages state return ast ncluded locks page page page pass ast construction first pass constructs syntax tree representing source pattern see details parsing specific pattern elements ast construction pass var input var nextsym sym lexer getsym sym type symtype eof null else lexer getline var scantosym symtype sym type symtype sym type symtype eof nextsym var hasownvalue obj var prop hasownproperty prop prop return true return false var getnatsym terminatorsym var sym type symtype nat type nodetype value sym value nextsym sym type symtype type nodetype natvariable value sym value nextsym else unexpected symbol scantosym terminatorsym return program parse parse cast parse pick parse body arse row parse row element parse parse undetermined repeat parse fixed repeat parse compound parse parse parse arse row repeat parse parse sample def parse sample parse parse condition parse var lexer input var sym nextsym sym type symtype eof addmsg msgtype warning pattern compile return else programparse sed page ncluded locks page page page page page page page page page page page page page page page page page page page page page pass sample substitution second pass responsible replacing sample call nodes children corresponding sample definition updating parameter values according parammap object sample call ast traversed depth first pass function samplesubstitutionpass starts traversing children pattern node empty object parammap sample substitution pass var pass children pass sample children pass update pass update condition var parammap type nodetype root pattern null parammap sed page ncluded locks page page page page row stitch constructs require expressions variables updated nodes children child nodes traversed pass traverse children var node parammap node var node var node switch type nodetype parammap break nodetype samplecall pass replace sample break nodetype branch updatecondition parammap dobranch parammap node return break nodetype row nodetype rowrep nodetype fixedstrep nodetype ustrep nodetype compst updateexpressions parammap parammap break nodetype knit nodetype nodetype knittbl nodetype purltbl nodetype knitbelow nodetype purlbelow nodetype nodetype slipkw nodetype slippw nodetype yarnover nodetype knitfb nodetype purlfb nodetype make nodetype makel nodetype maker nodetype knittog nodetype purltog nodetype ssk nodetype ssp nodetype psso updateexpressions parammap break nodetype var node var node var exprchildren parammap value node exprchildren exprchildren break default break sed sample substitution pass page ncluded locks page traversing children node find sample call need remove sample call node tree pass replace sample call var node var node var samplechildren getsamplecallchildren parammap node samplechildren samplechildren sed pass traverse children page however sample call node replaced must acquire deep copy children corresponding sample definition children set children sample call node traverse update expressions sample calls avoid naming conflicts localmap object created deep copy parent node parammap parammap object current node must expressions updated according local map localmap updated current node updated parammap passed traversing children sample call pass sample call children var getsamplecallchildren node parammap var localmap jquery extend true parammap var domainval node parammap node parammap domainval localmap localmap domainval node parammap domainval var sampledef jquery extend true samples node name node sampledef node localmap node sed sample substitution pass page two properties currently allow natural number expressions repcount num node either properties expression updated according passed parammap pass update expressions var updateexpressions node parammap node repcount node repcount type nodetype node repcount parammap node num node num type nodetype node num parammap sed sample substitution pass page pass pattern verification third pass traverses syntax tree determines structural issues pattern see details verification specific pattern elements verification pass var cast pick row row elem children node row elem var node node var node verifynode node var verifynode node node switch node type nodetype root verifynode node break nodetype node break nodetype node break nodetype caston verifycaston node break nodetype pickup verifypickup node break nodetype bindoff node break nodetype node break nodetype row verifyrow node break nodetype rowrep node repcount node break nodetype node break default break verifynode sed page ncluded locks page page page page page page page page page page code generation html back end purl compiler generates html code representing pattern given abstract syntax tree ast representing knitting pattern ast returns html display knitting pattern according knitting pattern standard pattern element usually contained within div corresponding class name also tags added pattern section names order elements pattern written seen passes compiler see details code generation specific pattern elements var patterntextwriterhtml code gen types code gen tag writing write html node write html root write html write html write html cast write html pick write html write html write html body write html row write html write html undetermined repeat write html fixed repeat write html compound write html row repeat return generate writeroot ncluded locks page page page page page page page page page page page page page page page page page div span tag types used class type associated pattern node output language used styling code gen types var classtype pattern pattern patternname patternname patternnote sectionname sectionname sectionnote caston caston castonnote castonnote pickup pickup body body row row rownote rownote rowrep rowrepeat stitchcount stitch bindoff bindoff bindoffnote bindoffnote join error warning warning verification var tagtype div div span span sed page following functions used code generation module create html tags code gen tag writing functions var addelement tag classtype openelement tag closeelement var openelement tag class var closeelement sed page function writenode first checks node messages error tag added result mark node message node haserrormsg addelement tagtype span classtype error node haswarningmsg addelement tagtype span classtype warning node addelement tagtype span classtype sed write html node page appropriate node writing function called based node type write html node var writenode node var node return mark node message switch node type nodetype root writeroot node break nodetype node break nodetype node break nodetype caston writeco node break nodetype pickup writepu node break nodetype bindoff writebo node break nodetype node break nodetype row writerow node break nodetype rowrep writerowrep node break nodetype fixedstrep writefixedstrep node break nodetype ustrep writeustrep node break nodetype compst writecompst node break nodetype knit nodetype nodetype knittbl nodetype purltbl nodetype knitbelow nodetype purlbelow nodetype nodetype slipkw nodetype slippw nodetype yarnover nodetype knitfb nodetype purlfb nodetype make nodetype makel nodetype maker nodetype knittog nodetype purltog nodetype ssk nodetype ssp nodetype psso node break default break return sed page ncluded locks page discussion language purl provides format reuse efficient storage knitting patterns compiler generating assembled formatted pattern use knitter present state project good foundation writing knitting patterns many desirable features still missing planned components removed initial implementation due time constraints desire give component time thought deserved proper implementation listed postponed features next steps project approximate order priority user defined requirements natural component knitting pattern samples construct concept requires research planning initially anticipated top priority next large feature test page test page provided gives idea language works practical purposes following necessary load pattern files compilation allow pattern compilation multiple files pattern samples would reusable multiple patterns save feature download target pattern pdf format charts originally compiler planned code generation patterns standard text notation chart notation charts written read style different text patterns column repeats well row repeats alternating rows requiring visual reverse written stitch performed large amount optimization would required make pattern concise purl output chart reasonable size small updates productive pattern design setting following would desirable display pattern attributes designer name date recommended yarn needles commenting pattern source file notes target pattern appendices appendix extra productions expressions expressions currently allow addition natural number literals variables hexpri hidenti hnati hidenti hnati expression node contains children objects type expression parse var terminatorsym var node type nodetype expression sym type symtype nat sym type symtype node push getnatsym terminatorsym sym type charsym plusop nextsym node push getnatsym terminatorsym node sed ast construction pass page expression verified adding values children natural number literals verification pass setting value expression node verify expression var node node value node var node node value node value sed verification pass page write expression var natobj var var natobj var natobj var natobj type nodetype natvar value type nodetype push value push condition hconditioni hexpri hexpri condition node includes property comparison operator node left node right operator condition parse var var node type nodetype condition sym type symtype nat sym type symtype node nodel switch sym type charsym equal node comparison comparetype equal nextsym break charsym openangle node comparison comparetype less nextsym break charsym closeangle node comparison comparetype nextsym break symtype lesseq node comparison comparetype lesseq nextsym break symtype greatereq node comparison comparetype greatereq nextsym break default break sym type symtype nat sym type symtype node noder node sed ast construction pass page pass update condition var updatecondition node parammap node nodel parammap node noder parammap node nodel value node nodel var node nodel node nodel value node nodel value node noder value node noder var node noder node noder value node noder value switch node comparison comparetype equal node dobranch node nodel value node noder value break comparetype less node dobranch node nodel value node noder value break comparetype lesseq node dobranch node nodel value node noder value break comparetype node dobranch node nodel value node noder value break comparetype greatereq node dobranch node nodel value node noder value break default node dobranch break sed sample substitution pass page appendix compiler types types used throughout compiler parser types var sidetype var cotype flat flat circular circular prov provisional var rowtype row row rnd rnd var colortype main contrast var yarnpostype fro wyif back wyib var nodetype root root pattern pattern section section caston pickup bindoff join row rowrep sampledef join row rowrepeat sampledef samplecall samplecall fixedstrep fixedstrep ustrep undeterminedstrep compst compst knit purl knittbl purltbl knitbelow kbelow purlbelow pbelow slip slipkw slippw yarnover knitfb kfb purlfb pfb make makel maker knittog purltog ssk ssk ssp ssp psso psso expression expr natliteral natlit natvar branch branch var msgtype error error message warning warning message verification verification message var comparetype equal less lesseq leq greatereq geq parts lock page page symbol lookup types var charsym comma period colon asterisk plusop minusop semicolon openparen closeparen openbrack closebrack openangle closeangle verticalbar equal var symtype nat nat ident ident string rowrep lesseq leq greatereq geq eof eof unknown unknown var keywordsym pattern pattern caston pickup bindoff join join castoncirc circular castonprov provisional section section sample sample last last end end row row rnd rnd repeat repeat yarninfront wyif yarninback wyib colormain var stitchsym knit purl knittbl purltbl knitbelow purlbelow slip slipkw slippw increases yarnover knitfb purlfb make makel maker decreases knittog purltog ssk ssp psso parts lock page page appendix examples tests number pattern examples tests provided web page tests intended show variety features available language exhaustive project display board pattern sample edging row end repeat sample row repeat project display board edging edging sed pattern tests page market bag pattern sample max max rnd rnd end max sample diagonallace rnd end rnd end repeat sample type type row end row end repeat type rnd end rnd end repeat market bag body rnd end rnd end diag onalla handle body top row row body top sed pattern tests page shawl pattern sample shawlrep row row row row sample shawlbody shawlrep shawlrep shawlbody shawl shawlbody sed pattern tests page basic stitches test basic sts row row row row row row row row row row row row ssk row ssp row psso row row kfb row pfb row row row sed pattern tests page compound stitch test compound stitch test row sed pattern tests page fixed stitch repeat test fixed stitch repeat test row sed pattern tests page undetermined stitch repeat test undetermined stitch repeat test row end row sed pattern tests page row repeat test row repeat test row end repeat sed pattern tests page section test section test first section row second section row sed pattern tests page sample test sample row repeat sample test sed pattern tests page sample branch test sample samplebranch row row branch test samplebranch samplebranch sed pattern tests page recursive sample test sample row end row end sample recursion sed pattern tests page row type test row type test rnd row sed pattern tests page color options test color test row row sed pattern tests page errors test error test row row row sed pattern tests page pattern tests div tests button shawl javascript loadtest div pattern example shawl button market bag javascript loadtest div pattern example market bag button project display board javascript loadtest div pattern example display board button basic stitches javascript loadtest div test button compound stitch javascript loadtest div test compound button fixed stitch rep javascript loadtest div test fixed repeat button undetermined stitch rep javascript loadtest div test undetermined repeat button row repeat javascript loadtest div test row repeat button sections javascript loadtest div test button sample javascript loadtest div test sample button sample recursion javascript loadtest div test sample button sample branch javascript loadtest div test sample branch button row type javascript loadtest div test row type button color options javascript loadtest div test color options button errors javascript loadtest div test sed page ncluded locks page page page page page page page page page page page page page page page appendix test page dom doctype html html head stylesheet styles stylesheet styles pattern javascript http code jquery javascript javascript lexer javascript parser javascript codegen script function loadtest idstr pattern showmessages msgarr var msgarr var msgobj msgarr var msg msgobj sectionname msgobj sectionname msg push section msgobj sectionname msgobj rowindex msg push row msgobj rowindex var msgobj msg push line msgobj num msgobj pos msgobj charpos msg push msgobj message var class msgobj messagetype var value var onclick movecursor cursorpos var input type message display append movecursor pos document getelementbyid pattern pos document getelementbyid pattern pos pattern document ready compile input pattern display empty message display empty var parse pattern messages showmessages messages var output patterntextwriterhtml generate pattern display prepend output body div header div left content div left wrapper div entry wrapper pattern false example pattern section pattern row row row end repeat div compile input button compile div message display message div center content div center wrapper div weave div weave bottom div tests wrapper div right content div right wrapper div pattern display div display ast display div display ast display div display ast display div footer ncluded locks page appendix style sheets test page styles margin padding border border html body html knitting header position absolute top left right width padding width padding width display block width padding convergence cin shadows shadowsintolighttwo regular sofia regular entry position relative compile position absolute bottom width compile input input white border compile input hover input hover compile input input compile input display table margin auto padding width overlockblackit width dbdbdb border cccccc overflow auto input white normal width border red border yellow border purple center padding width margin auto padding overflow webkit none put width padding white normal hidden none right width overflow auto padding white border cccccc position absolute width border position absolute width bottom border fonts artifica regular delius regular novacut novacut novaslim novaslim radley italic convergence convergence regular overlockreg regular overlockblackit blackitalic ttf overlockit italic overlockblack overlockbold novaround novaround target language styles patternname sectionname caston pickup margin body margin row margin rowrepeat margin rowrepeat stitchcount warning bold purple red warning yellow white none hidden bibliography craft yarn council standards guidelines crochet knitting http accessed eclipse standard https version kepler service release build ravi sethi jeffrey ullman alfred aho monica lam compilers principles techniques tools prentice hall therese dillmont encyclopedia needlework mulhouse alsace available free project gutenberg seffen ernst literate programming eclipse http version donald knuth literate programming computer journal richard rutt history hand knitting batsford laura spradin grrlfriend market bag http knitting pattern accessed lesley stanfield melody griffiths essential stitch collection reader digest association
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computing total displacement number via weighted motzkin paths andreas barbara daniel tomas paolo thomas jun eth department computer science abstract counting number permutations given total displacement equivalent counting weighted motzkin paths given area petersen former combinatorial problem still open work show connection allows construct efficient algorithms counting sampling permutations algorithms provide tool better understand original combinatorial problem approach different way counting based certain building sequences motzkin paths may independent interest introduction consider set permutations elements diaconis graham studied disarray statistic permutations also called total displacement knuth problem defined follows permutation define distance identity permutation sum displacements elements note distance always even following natural question still unresolved many permutations given distance identity permutation one would like know following total displacement number number permutations total displacement equal far closed formula arbitrary known fixed seven entry oeis reports values small introduction figure permutation motzkin path width area petersen made recently significant progress question showing permutations correspondence motzkin paths whose area exactly distance consideration result shows motzkin path see area one easily calculate number permutations correspond specific path therefore problem translates problem counting weighted motzkin paths given area motzkin path consists sequence downright moves lattice starting coordinate path never goes axis ends axis see figure right example path one consider width area defined number moves area region axis path permutations elements total displacement map motzkin paths width area instance permutation figure mapped motzkin path according following rule first element mapped element position goes higher position right also number coming position higher fourth element mapped opposite happens finally elements mapped neither previous cases apply let denote maximum height path move move move anytime number permutations map certain motzkin path also refer weight example figure gives note formula separates moves motzkin path independence exploit article theorem let set motzkin paths width area holds corollary appendix given motzkin path length sample uniformly random one many permutations mapping time contribution work address counting sampling permutations combinatorial computational point view specifically computational side show total displacement number computed efficiently namely time space introduction combinatorial side introduce sequences certain building blocks provide different perspective problem structure moreover crucial part markov chain sampling method constitutes third contribution paper finally consider task sampling permutations given total displacement uniform distribution compute number permutations efficiently look paths left right building operation introduced barcucci provide elegant dynamic programming formulation achieves running time needs space consequently compute sequences much higher values possible considering combinatorial aspects show every motzkin path comes sequence describing building blocks provide explicit formula number paths building blocks form weights equation preserved sense weight path depends building sequence since exact formula seems currently reach achieve good estimates large contribute sampling methods also independent interest particular dynamic programming algorithm provides sampler complexity algorithm show sampling sequences building blocks appropriate distribution automatically gives sampler permutations one application latter result monte carlo markov chain mcmc method gives alternative approach dynamic programming computational experiments mcmc method show promising convergence speed leading sampler high values experimental results support hypothesis mcmc method faster method based dynamic programming runs time related work different metrics permutations studied survey see sampling counting permutations fixed distance studied several metrics total displacement number motzkin paths various conditions also studied general frame enumeration lattice paths motzkin numbers play role many combinatorial problems illustrated example total area set generalized motzkin paths horizontal segments constant length studied moreover author studies moments generalized motzkin paths first moment describes area motzkin path heinz describes different algorithm enumerating unweighted motzkin paths given area remark section use markov chains sampling counting combinatorial objects active research area see book works exploit connection combinatorial structures paths certain type accomplish task see paper organization section describes dynamic programming algorithm section describes weighted motzkin paths counted via building block sequences section provides markov chain sampling algorithm well experimental evaluation weighted motzkin paths using dynamic programming weighted motzkin paths using dynamic programming recall denote number permutations elements total displacement oeis let denote number motzkin paths width area oeis dynamic program counting weighted motzkin paths theorem computing done time space proof key ingredient construction barcucci produces every possible motzkin path unique sequence insertion steps let look last fall given motzkin path suffix moves one positions fall moves insert new peak insert new flat repeatedly inserting peaks flats way along last fall create path see figure example construction complete unique meaning every motzkin path created unique sequence insertions allows derive dynamic programming formulation counting add last fall length state write recursively express undo last insertion step inserted flat last insertion last fall least long insert inserting peak might increase last fall length one also possible predecessor states together base case gives recurrence allows many states hence immediately get time algorithm space shave one factor time space follows first note compute two sums constant time precompute prefix sums last variable let denote prefix sums allows compute every value amortized constant time time overall finally recurrence relies last two values computing many values figure six possible last fall length weighted motzkin paths using dynamic programming need stored values simply marginals last fall lengths extend recurrence weighted case corollary gives rise total displacement count distribute factors weight equation steps dynamic program denotes height last flat peak add factors remark bounds theorem assumed basic operations numbers involved exponential however easily bound showing length log dynamic programs use multiplication small numbers size log addition one consider refined analysis multiplying time space bounds theorem polylog finally suggested anonymous reviewer space could improved counting modulo small primes using chinese reminder theorem remark computing oeis contains dynamic program heinz stated maple code snippet without comment reference uses different state might time complexity believe extension weights also applied sampling dynamic program theorem running dynamic program theorem sample weighted motzkin paths time proof given access source randomness filled table randomly retrace steps dynamic programming states sample motzkin path right left weighted paths according steps exactly first sample last fall length picking random number finding smallest prefix sum larger continue offset within class paths last fall length step first decide whether recurrence comparing left summand know whether move last fall increment find last fall length previous state adapt recurse end note search initial takes linear time every time compare value fix least one move sampled motzkin path sampling takes time overall remark sampling procedure requires full table dynamic program stored hence memory optimization theorem used simultaneously remark implementation counting sampling approaches theorems available http people inf ethz code quickly compute integer sequences known combinatorial structure motzkin paths figure motzkin path left obtained building blocks combinatorial structure motzkin paths section look combinatorial structure motzkin paths natural decomposition motzkin path building blocks already hinted last section height motzkin path count number flats peaks definition building sequence given positive integers finite sequence nonnegative integers building sequence following two conditions hold hfh set building sequences satisfying denoted sequence natural interpretation set building blocks generate number motzkin paths width area see figure flats peaks height split pieces width rearranged motzkin path proposition motzkin path width area exists unique building sequence obtained splitting rearranging blocks sequence theorem gives surjective mapping permutations motzkin paths easy see number permutations mapping path given equation uniquely determined building block sequence since perm hence independent actual motzkin path depends combinatorial structure raises question whether also number motzkin paths share common building sequence solely determined answer positive deriving formula number denoted proceed fashion looking number peaks flats highest level rearranged level fixed proceed recursively arranging blocks one level theorem building sequence number motzkin paths width area constructed building sequence exactly combinatorial structure motzkin paths figure top construction paths given sequence note thus valley inserted height proof start highest flats sequence flats two flats either lie directly next might separated move followed point move call setting valley get valleys splitting peaks height reassembling way round see figure feasible motzkin path slope left slope right height pieces remaining valleys freely placed around flats choose places available positions number ways continue second highest level naturally number times motzkin path rises level exactly number peaks height distribute flats height around peaks pick many positions hence choose many possibilities placing flats place new valleys next lower level around existing peaks flats leftmost slopes fixed hence number ways distribute valleys given third factor equation since choices different levels independent iterate reasoning include flats height conclude corollary theorems corollary exists surjective mapping permutations elements building sequences satisfying following condition building sequence number permutations distance identity permutation mapped building sequence precisely perm given equation perm equation therefore total number permutations distance identity permutation satisfies sampling weighted motzkin paths length area example building blocks figure yield motzkin paths path corresponds permutations permutations mapping building sequence remark theorem corollary allow dynamic program counting sampling weighted motzkin paths similar sections additionally easily sample paths fixed number highest peaks flats cost additional running time see appendix sampling weighted motzkin paths length area section consider task selecting sampling permutations uniform distribution permutations given total displacement corollary enough sample motzkin paths proper weights already seen section sample weighted motzkin paths using dynamic programming cost large memory consumption show section approach sample weighted motzkin paths based building sequences introduced section requires memory general observe sampling permutations accomplished efficiently sample building sequences probability proportional perm polynomial time theorem every algorithm samples sequences probability turned algorithm sampling permutations uniformly random among permutations elements total displacement proof given sequence sampler maps sequence random motzkin path random permutation follows pick motzkin path uniformly random among created probability pick permutation among map motzkin path probability perm step sequences motzkin paths construction used prove theorem suggests also sample one motzkin paths given sequence uniform distribution namely pick positions valleys height uniformly random equation pick positions flats uniformly random equation repeat lower level exactly described construction since particular path corresponds exactly one choice steps equation probability precisely step paths permutations shown corollary preliminary definitions markov chains section introduce definitions markov chains used throughout work see markov chain finite state space specified transition matrix probability moving state state one step tth power transition matrix gives probability moving one state another state steps chain studied work ergodic see proof meaning unique stationary distribution two states sampling weighted motzkin paths length area reversible chains shall use definition reversible markov chain also called detailed balanced condition transition matrix admits vector stationary distribution chain transitions total variation distance mixing time total variation distance two distributions mixing time ergodic markov chain transition matrix defined tmix min common also define quantity tmix tmix justified fact tmix tmix experiments shall evaluate total variation distance get better estimate markov chain sampler suppose set possible local changes transforming sequence another sequence sequences obtained applying certain number operations following standard metropolis chain samples sequences desired distribution probability nothing otherwise select one local operations operation applied current sequence new sequence unfeasible nothing otherwise let sequence obtained applying operation accept operation transforming probability perm min min perm nothing remaining probability local operations sequences define metropolis chain mblocks four types operations peak flat flat valley flat flat peak valley formally define sampling weighted motzkin paths length area figure left middle canonic path resp right building canonic path red given path black area note type operation applies two indices also implicitly consider reversed operations undo changes explain step chain mblocks detail markov chain mblocks picks two indices random picks one four operations decides probability whether choose operation reversed version step computing transitional probability done constant time factors equations change theorem markov chain mblocks defined ergodic unique stationary distribution satisfies every proof proof consists two steps first show chain ergodic aperiodic connected see use standard detail balance condition obtain stationary distribution connectivity mblocks prove chain mblocks connected every building sequence reach every building sequence maximum operations argue two steps intuitively show transform two paths operations depicted figure seen every operation figure corresponds sequence operations markov chain mblocks given figure formally every path width area turned path area width using operations figure prove consider canonic path given width area canonic path uniquely defined path following holds every steps maximum area among paths possible forms canonic path shown figure given path width area transformed canonic path area using steps figure overlay given path canonic path proceed steps right schematically shown figure black path given path red path canonic path three possibilities either paths coincide case proceed right given path differs proceeding move move cases given path must intersect canonic path falling part otherwise area use operations figure horizontal sweeps left right missing area canonic path end paths must coincide areas operations simulated operations sequences figure seen immediately four cases figure correspond directly one two operations figure sampling weighted motzkin paths length area peak flat flat valley flat flat peak valley figure basic operations sequences operations figure changing path shape one side transformed shape side cases need two operations defined figure sampling weighted motzkin paths length area stationary distribution mblocks metropolis chains acceptance probability min stationary distribution since detailed balance condition obviously satisfied given number operations definition yields detailed balance condition experimental evaluation mblocks interested required number steps distribution mblocks sufficiently close stationary distribution measure distance two distributions total variation distance mixing time markov chain smallest time total variation distance stationary distribution distribution steps starting state smaller small study mixing time mblocks given area given width running following experiment estimate distribution given number steps repeatedly running mblocks initial state defined follows building block sequence consists one peak height every remaining area width filled greedily flats maximal possible height total variation distance distribution mblocks number steps stationary distribution estimate mixing time given area given width computing total variation distance increasing total variation distance figure left illustrates mixing time width every area one possible building block sequence maximal mixing time steps necessary area fact every width smaller mixing time maximal area even otherwise due choice initial state mblocks estimate maximal mixing time widths larger computing mixing time number repeats necessary estimate distribution mblocks steps depends number possible building block sequences grows exponentially depending figure right shows maximal mixing time width plot suggests number steps necessary approximate stationary distribution grow exponentially depending width algorithm probably faster sampler based dynamic programming results suggest mcmc sampler achieves mixing time conjecture mblocks mixes time remark implementation mblocks available http people inf ethz area area area area area area area area area area area area area mixing time total variation references time figure left total variation right maximal mixing time given widths mixing time areas references elena barcucci alberto del lungo elisa pergola renzo pinzani construction enumerating motzkin paths computing combinatorics pages springer russ bubley randomized algorithms approximation generation counting springer emeric deutsch alois heinz motzkin paths area online encyclopedia integer sequences http june michael deza tayuan huang metrics permutations survey journal combinatorics information system sciences persi diaconis ronald graham spearman footrule measure disarray journal royal statistical society series methodological pages robert donaghey louis shapiro motzkin numbers journal combinatorial theory series olivier mathieu alois heinz permutation fixed total displacement online encyclopedia integer sequences https may ian goulden david jackson combinatorial enumeration dover publications sam greenberg amanda pascoe dana randall sampling biased lattice configurations using exponential metrics symposium discrete algorithms soda pages references width area runs mixing time relevant areas table experimental setup distribution steps estimated using runs mblocks maximal mixing time estimated computing mixing time widths larger mathieu kyle petersen generating function total displacement electronic journal combinatorics katherine humphreys history survey lattice path enumeration journal statistical planning inference ekhine irurozki sampling learning probability models permutation spaces phd thesis university basque country donostia san july donald knuth art computer programming sorting searching david asher levin yuval peres elizabeth lee wilmer markov chains mixing times american mathematical donatella merlini generating functions area lattice paths discrete random walks drw pages pergola pinzani rinaldi sulanke bijective approach area generalized motzkin paths advances applied mathematics robert sulanke moments generalized motzkin paths journal integer sequences postponed proofs appendix postponed proofs corollary appendix given motzkin path length sample uniformly random one many permutations mapping time proof given motzkin path sample permutation among permutations map consider feasible sequence letters denoting diagonally moves diagonally moves moves following scan sequence left right new found match left yet matched choose step constructs edges scan right left sequence matching newly encountered right yet matched step matchings previous step count step symmetric previous one constructs edges choose fixed point edge meaning following number edges crossing corresponding flat height equal number edges also crossing property due balanced matchings options options correspond breaking one edges one edges last option let map trivial cycle permutation choose one options experiments estimate distribution given number steps repeatedly running mblocks appropriate number steps depending width table compute total variation distance distribution mblocks steps stationary distribution need know building block sequences let set building block sequences reached mblocks steps least one run total variation distance steps perm experiments mixing time mixing time maximal mixing time maximal mixing time width width figure plot maximal mixing time given width filled circle mixing times area cross linear left logarithmic right dashed line right plot shows total variation width total variation width area area area area area area total variation total variation area area area time time figure total variation width left respectively right every area experiments area area area area area area area area area area area area area total variation total variation area area area area area area area area area total variation width total variation width time time figure total variation width left respectively right every area time total variation total variation area area area area area area area area area area area area area area area area area total variation width total variation width area area area area area area area area area area area area area area area area area area area area area area time figure total variation width left respectively right every area dynamic programming approach total variation total variation total variation width total variation width time time figure total variation width left respectively right every area dynamic programming approach sake clarity first describe dynamic programming procedure counting unweighted motzkin paths given width area specifically define following subproblem number motzkin paths width area height note forbid flats height fix number peaks height count number paths given height including flats height dropping condition fixing number peaks height simply compute number motzkin paths given width area obtained summing also sum possible areas get classic motzkin numbers next show computed polynomial time make use equation theorem number motzkin paths constructed building sequence though task solved easier dynamic programming approach extend weighted case main goal dynamic programming approach theorem table computed many possible parameters total time proof recall forbid flats height fix number peaks height allows following recursive counting base case parameters equal else recursion otherwise parameters strictly positive recursive case enumerate potential numbers flats peaks height look subproblems subproblems weighted derived equations number possible interleavings thus shown table computed many possible parameter values time bottleneck two nested sums equation let consider problem counting weighted motzkin paths function equation end extend definition count path according weight perm resulting table computed recursively fashion incorporating recursion two terms defining perm equation peak options flat options base case identical unweighted case equation drop condition flats last level count weighted motzkin paths given area width simply time computing overall space asymptotically used thus proven following corollary table marginals computed many possible parameter values total time
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cloud edge computing mobile services decision making nov minimize energy consumption meysam masoudi student member ieee cicek cavdar member ieee abstract promising technique provide mobile applications high computation resources offload processing task cloud mobile cloud computing enables mobile devices limited batteries run resource hungry applications help abundant processing capabilities clouds save power however always true cloud computing consumes less energy compared mobile edge computing may take energy mobile device transmit file cloud running task edge paper investigates power minimization problem mobile devices data offloading ofdma mobile cloud computing networks consider maximum acceptable delay tolerable interference qos metrics satisfied network formulate problem mixed integer nonlinear problem converted convex form using approximation solve optimization problem proposed centralized distributed algorithms joint power allocation channel assignment together decision making simulation results illustrate utilizing proposed algorithms considerable power saving could achieved short delays large bitstream sizes comparison baselines index terms offloading resource allocation mobile cloud computing mobile edge computing ntroduction swift growth development resource hungry mobile applications motivated users use smart phones platform running applications however mobile devices part work accepted ieee study supported celtic plus project soogreen service oriented optimization green mobile networks always considered platform resource hungry applications due limited power processing capacity moreover one key concerns users battery lifetime mobile devices running applications knowing fact increasing clock frequency cpu increases power consumption therefore tension resource hungry applications resource poor mobile devices tackle aforementioned problem one solution bridge gap available required resources offloading burden mobile devices cloud cloud computing abundant processing resources become attractive solution order ease pain storage data processing cloud computing mobile applications enable new services mobile users true cloud computing potentially save energy mobile users however always true device consumes energy transmit data cloud process data interference radio channel conditions transmission data may consume energy mobile device however trivial decide making simple comparison two energy figures device served one base station since decision may create interference change channel conditions neighboring devices surrounding cells also another important factor impact decision delay decision making procedure must consider delay sensitivity applications determine whether choose local processing offloading mobile devices consume joules per bit delay requirement gets stringent process certain task delay requirements different mobile broadband services seen table paper investigate energy saving potential data offloading mobile devices scenario propose efficient algorithms make decisions simultaneously mobile devices minimize total energy consumption meeting delay requirements services channel assignment power allocation problems considered jointly related works mobile cloud computing mcc provides infrastructure platform software services mobile users hand interaction cloud mobile user inevitable mcc consequently users decide offload data cloud necessary efficiently utilize available resources otherwise users benefit potential advantages mcc words resource management schemes key techniques table acceptable delay different services service type acceptable delay online games omnipresent third person avatar first person avatar audio services voice video services video data non services seconds guarantee quality service qos mcc networks conducted surveys addressed existing studies integrating mobile edge computing mec mobile networks computation offloading schemes resource management problems current challenges accordingly main focus model energy consumption applications mcc networks authors also proposed energy aware resource allocation algorithm scheduling cloud framework offloading computation cloud proposed investigated offloading infrastructure eased migration code cloud main goal study mobile code offloading architecture illustrated significant energy saving obtained using offloading methods task offloading different applications one user case studied using experimental measurements shown wireless access inevitable effect performance mcc authors considered problem resource scheduling mcc networks also heuristic approach adopted minimize energy consumption users making decision offloading resource allocation task authors modeled energy consumption mobile devices formulated optimization problem minimize energy consumption single device data offloading dynamic application task offloading algorithm using lyapunov optimization proposed aiming minimizing energy consumption users constraint maximum acceptable delay application authors presented practical offloading framework cost aware system considering trade offs game theoretic approach adopted design offloading mechanism mobile devices model case considered corresponding qos well effect users ignored decentralized offloading game proposed make decision among mobile devices simple single channel scenario partitioning problem mobile data stream application defined used genetic algorithm solve problem also reported partitioning data enhance application performance terms throughput authors utilized markov decision process approach solve problem task offloading formulated delay minimization problem find optimal task scheduling policy authors studied problem network energy minimization based mcc system study authors jointly optimized beamforming design power allocation decision making strategy energy consumption latency minimization problem partial computation offloading algorithm optimize computational speed mobile devices transmit power proposed authors deal latency issue means cloudlet infrastructure data center bring cloud closer users authors proposed model mobile device energy consumption also derived offloading policy considering delay energy consumption single stochastic wireless channel good bad channel state model limited singleuser case interference users qos addressed model authors considered simple computing system proposed algorithm optimize power consumption minimize delay study interference analysis effect offloading decision missing authors solved offloading optimization problem remove processing burden mobile devices without considering resource allocation authors modeled offloading decision competitive game users try minimize energy consumptions consider power allocation significant impact performance algorithms minimize offloading energy consumption authors proposed joint optimization computing radio resources considering latency constraints computing network proposed joint power allocation decision making channel assignment algorithm perform resource allocation considering interference delay constraints contributions still plenty challenges tackled multichannel mcc networks best knowledge problem resource allocation decision making data offloading multi cell networks considering multi users addressed literature paper aim minimizing power consumption users considering user qos terms delay maximum tolerable interference channel formulate resource allocation offloading optimization problem show problem mixed integer nonlinear problem minlp optimal solution intractable tractable solution convert problem convex form propose two algorithms called solve problem polynomial time main contribution paper summarized follows context multi cellular multi user ofdma mcc networks formulate resource allocation offloading problem aware network status users demand aiming minimizing total power consumption users subject constraints qos users interference threshold formulated problem mixed integer nonlinear optimization problem minlp solve problem converted convex form using variable changing approximation adding penalty factor relaxing binary constraints therefore problem solved polynomial time also propose two algorithms solve problem resource allocation decision making first algorithm centralized scheme designed performed base station second one distributed scheme requires partial information exchange suitable performed user terminal complexity algorithms also investigated simulations show exists offloading region user offloading help save power comparing cell edge user normal user network show optimal region depends delay threshold bit stream size users also position channel condition users rest paper organized follows section system model presented problem formulation solution methodology discussed section propose mobile device mobile device mobile device server mobile device server mobile device mobile device mobile device mobile device mobile device mobile device server mobile device mobile device mobile device server mobile device mobile device mobile device mobile device mobile device mobile device mobile device server mobile device server mobile device mobile device mobile device server mobile device mobile device mobile device mobile device server mobile device mobile device server mobile device server mobile device mobile device mobile device mobile device mobile device server mobile device server mobile device mobile device mobile device mobile device mobile device mobile device mobile device server mobile device mobile device mobile device fig system model algorithms corresponding complexity analysis section followed simulation results presented section finally bring concluding remarks section ystem odel roblem ormulation system description according consider cellular network base stations mobile users mus uniformly distributed within cell range base station equipped server responsible offloaded users data processing assume centralized unit exchanges required information base stations using backhaul cell serve active users assume available bandwidth divided subchannels model adopted composed large scale fading small scale fading shadow fading also consider ofdma access method hence users cell share however user might experience interference neighboring cells model user cell bit stream size generated users bit stream size normal distribution mean variance users process data send cloud users use schemes sending portion data cloud processing remaining data locally data corresponds user cell processed within maximum acceptable delay delay threshold generated normal distribution mean variance assume processed data short response time delay neglected power model local processing power model users supposed process data locally cpu power consumption dominant composed dynamic power circuit power leakage power dynamic power dominating power cpu function required cpu cycles depends delay threshold input data size optimal value cpu frequency minimum power consumption cpu proportional maximum acceptable delay users bit stream size scaling factor power consequently use following model local processing power consumption plocal lni plocal local processing power consumption user cell constant value depends cpu application parameters offloading power model transmission power sending data cloud ptx pti denotes transmission power consumption user cell binary variable representing whether corresponding assigned user therefore user total transmission power ptx power amplifier coefficient constant circuit power aggregated power model total power consumption active users network written otal pti plocal mlni integer variable takes value user cell uses processor takes value user sends data cloud therefore total power consumption written otal mlni moreover signal noise plus interference ratio base station cell given ptx channel gain jth ith cell denoted channel gain user cell cell denoted hik first term denominator noise power second term interference cells channel cell calculated ptx assumption users must utilize whole duration considering fixed power minimization line energy minimization problem formulation section develop mathematical formulation decision making resource allocation problem base station determines offloading users allocates users specifies suitable power level objective resource allocation minimize aggregated power consumption users allocating resources offloading users way delay requirement satisfied optimization problem formulated follows otal min subject ptx pmax nhik ith tmax roc rmax objective minimize total power consumption active mus network constraint indicates transmit power user limited pmax constraint states base station interference arising cells restricted within threshold constraint restricts maximum tolerable delay user cell tmax aforementioned user sends data cloud user decides process data locally cpu responsible satisfying constraint analysis assume cpu uses entire available time reduce power consumption constraint addresses processing limitation cloud constraint guarantees ofdma assumption cell assigned one user constraints indicate data offloading indices binary variables worth mentioning constraint written equivalent form using tmax defining rmin tmax noting left side total data rate user cell obtain following equivalent constraint rmin rest paper consider constraint form presented optimization problem defined mixed integer nonlinear problem minlp finding optimal solution solved polynomial time nonconvexity coming three reasons problem first second reasons binary inherent decision making variable constraint combinatorial nature allocation constraint third one due constraints existence power allocation variable denominator sinr formula defined following section address deal variables solve problem converting convex form iii olution ethodology section aim transform primary problem defined section canonical convex form regard classify challenges two categories binary variables non convex functions resolve challenges caused binary variables one approach relax troublesome constraints allocation instance shape problem convex form making hard decision end alternative approach add auxiliary constraints enforce solution desired form describe later another approach break problem one could successively first solve problem annoying binary variable consequently given variable solve rest problem deal functions utilize theory optimization superclass convex functions called difference convex functions later demonstrate problem written form functions end applying taylor approximation enables solve last stage converting primary problem defined convex form powerful approaches available tackle problem follows first step break problem two solve successively first determine channel assignment user cell second find decision variable power allocation use solution first input second also results second used update solution first process continues convergence furthermore apply two approaches solve second subproblem overview two utilized approaches solve problem seen equations first approach separating assignment problem solved jointly variables power allocation decision variable follows iteration initialization iteration optimal solution second approach separate assignment power allocation decision variable follows iteration initialization iteration optimal solution main difference two approaches former jointly solve problem power allocation decision making however latter divide second two steps solve individually following subsections first deal solving first followed solving second converting convex one optimal assignment given power allocation vector optimal assignment power allocation offloading next iteration follows proposition given power vector minimum power consumption attained cell assigned highest effective interference proof problem power minimization also minimum data rate requirement users satisfied minimum power consumed inequality minimum required rate becomes equality let assume users given best possible channel reach data rate minimum power consumption also let user channel effective interference value lower highest value user data rate rmin channel thus consumed power channel rmin constant value also assumption know effective interference denominator highest possible value hence assign highest effective interference value user total power consumption lower contrast assumption minimum power consumption therefore minimum power consumed maximum effective interference criterion channel allocation words higher effective interference less power consumed satisfy minimum required rate let eii denotes effective interference vector user channel high effective interference channel means experiencing good channel condition low interference cells therefore decision channel allocation made based following criterion eii thus channel allocation matrix time formed elements obtained equation stage solved first results available next steps next two subsections solve second introduced two power allocation decision making previous subsection solved problem assignment therefore one challenges primary problem resolved results previous subsection used section solve power allocation decision making two approaches applied tackle challenges approaches discussed following subsections joint power allocation decision making given assignment problem joint power allocation data offloading rewritten minp otal subject pti pmax nhik ith rmin roc rmax solve first reformulate mathematically tractable form since binary variable write moreover problem consists product terms use following change variable recast optimization problem also optimization problem includes integer variable hence convert continuous variables express constraint intersection following regions hence write optimization problem follows min otal problem continuous optimization problem respect variables however aim find integer solutions attain goal add penalty function objective function penalize values integer thus problem modified min lagrangian defined otal penalty factor shown sufficiently large values optimization problem equivalent attains optimal value proposition sufficiently large values optimization problem equivalent proof start point optimization problem expressed min max dual problem written max min suppose denote optimal solution optimal value optimization problem respectively min max max max min min max problem recall words increasing function according bounded optimal value problem feasible main problem result max min max comparing conclude strong duality holds max since monotonically increasing function respect otal min max problem optimal point second case goes monotonicity respect contradicts inequality states bounded thus term zero results first case hold optimization problem converted following problem min subject pmax hik ith nhi rmin roc rmax hik write objective function pnc two convex functions similar way define nhi hik write constraints follows rmin roc concave functions therefore problem form difference two convex concave functions programming programming start feasible initial point iteratively solve optimization problem let denote iteration number iteration make problem convex using first order taylor approximation follows qti zit solutions problem iteration denotes gradient operation respect thus iteration instead dealing problem solve following convex problem min subject rmin roc shown programming results sequence feasible solutions iteratively achieves better solutions previous iteration converges proposition programming results sequence feasible solutions iteratively decrease total power consumption network proof show solutions feasible original problem first notice solution approximated problem iteration must satisfy constraint qti rmin zit roc hand since two concave functions respect due first order condition concave functions qti substituting results qti conclude qti rmin zit roc thus solution approximated problem feasible original problem show total power consumption network decrease iteratively since convex function due first order condition convex functions using considering fact objective function written iteration min thus total power consumption network decreases iterations continue channel assignment power allocation decision making similar subsection assume channel assignment vector given based proposition given assignment optimization problem rewritten minp otal subject pti pmax nhik ith rmin roc rmax applying method used previous section formulate problem programming optimization problem words similar rmin roc applying first order taylor approximation optimization problem written minp otal subject pti pmax nhik ith qti rmin zit roc given assignment power consumption vectors offloading decisions made users recall power consumption user cell user compare offloading local processing power consumption make decision follows local local lgorithm esign section based solutions propose two tractable algorithms solve optimization problem polynomial time first algorithm fits well situation information cells available centralized unit base stations charge performing offloading algorithms second algorithm suits well offloading algorithm performed mus sides partial information exchange required base stations algorithm algorithm performs joint power allocation decision making called designed solve convex optimization problem presented key idea make decision allocate power simultaneously channels assigned beforehand algorithm represents procedure solving optimization problem using algorithm algorithm joint power allocation decision making algorithm initialize power imax counter counter imax channel allocation calculate eii based form based eii power allocation offloading decision solve problem using interior point method update power vector based solution update according solution end update counter counter centralized unit updates based sends value back base stations end algorithm composed two main sections channel assignment based equation power allocation offloading decision performing second part power vectors offloading decisions updated base station sent centralized unit centralized unit updates interference value channel sends back base station next iteration problem solved base station offloading algorithm performed besides plays important role performance algorithm penalty factor punish objective function value offloading decision variables equal therefore large enough penalize objective one fix value predetermined high value first set relatively low value case value real value next iterations tighten condition choosing larger algorithm section propose alternative algorithm less complexity decision making process moved mus side instead situation mus need partial information cells avoid integer inherent problem assign channels make offloading decision iteratively allocating power hence divide algorithm three main parts channel allocation done based offloading decision performed comparing alternative solutions power consumption according power allocation latter part channel allocation offloading decisions optimization variables anymore known user beforehand therefore algorithm performs channel allocation power allocation decision making iteratively called algorithm procedure finding solution algorithm presented algorithm algorithm channel allocation scheme algorithm offloading section user compares power consumption two possible cases local processing offloading makes decision accordingly given variables problem power minimization solved segmentation enables perform algorithm users side words second algorithm distributed scheme low data exchange requirements expense losing optimality centralized unit sends information interference base station base stations relay information users afterwards users use local information make decisions procedure continue convergence criteria met computational complexity proposed algorithms discussed compared next section complexity analysis section investigate computational complexity proposed algorithms assign users find user highest effective interference let denote maximum number users existing cell algorithm channel allocation power allocation decision making algorithm initialize initial points imax counter counter imax channel allocation calculate eii based form based eii offloading decision determine offloading decision based user update channel allocation offloading decision vector power allocation solve problem given channel allocation offloading decision vector using interior point method update power vector based solution found end counter counter centralized unit updates parameter based sends value back base stations base stations distribute users end max since finding maximum set elements requires operations assignment phase complexity order data offloading power allocation algorithm totally decision variables convex linear constraints therefore computational complexity solving joint data offloading power allocation problem given ncf algorithm data offloading power allocation separated find data offloading strategy sufficient compare power consumption cases user uses processor sends data cloud select one lowest power consumption since carry users cells need operations power allocation totally variables linear convex constraints similar presented first approach power allocation computational complexity order computational complexity proposed methods summarized table table computational complexity proposed approaches data offloading power allocation assignment imulation esults system parameters section evaluate performance proposed algorithms using numerical studies defining system parameters base line cases scenario depicted mobile network base station equipped computing server simulation parameters corresponding values summarized table iii assume cell serve users qos defined maximum acceptable delay carrier frequency set thermal noise considered zero mean gaussian random variable variance power spectral density pathloss model adopted shadow fading modeled zero mean log normal distribution variance rayleigh fading modeled exponential distribution cell coverage radius users distributed uniformly within cell coverage compared results two baseline cases understand main reasons behind power savings whether saving dominated offloading decisions stems power control channel first one mus use local processing nobody offloads data cloud comparing scheme observe much power saving obtained utilizing proposed algorithms second base line equal power allocation scheme power equally allocated user assigned channels required qos satisfied according given power allocation channel assignment performed based comparing scheme find amount power saving related power adjustment channel table iii simulation parameters values definition notation value bandwidth khz number number cells number active mus circuit power power amplifier efficiency scaling factor power maximum allowable interference ith dbm noise power spectral density maximum transmit power users pmax dbm maximum delay user cell average bit stream size user cell bits power consumption local computing bit stream size bits fig aggregate power consumption different bit stream sizes simulation results total power consumption users different bit stream sizes depicted larger bit stream size power consumed meanwhile gap local computing proposed algorithms power consumption increases illustrates could help users offload much power power saving percentage local user percentage bit stream size bits fig power saving percentage local computing users power consumption local computing maximum acceptable delay fig aggregate power consumption different acceptable delays saved seen figure increasing bit stream size percentage local computing users decreases reason local processing large bit stream size results higher power consumption comparison sending data cloud therefore confirmed simulations depicted users tend use alternative option offloading save energy large bit stream sizes using users decided local processing power saving attained comparison local computing base line comparing slightly outperforms terms energy saving complexity maximum acceptable delay quality service requirement another parameter affects power consumption offloading decisions investigated power saving percentage local user percentage maximum acceptable delay fig power saving percentage local computing users different acceptable delays delay impact algorithms longer acceptable delay offloading users means lower data rate requirement consequently lower power consumption sending data also local computing users results lower power consumption confirmed power model gap proposed algorithms benchmark wide beginning becomes tighter maximum acceptable delay gets longer discover illustrated percentage local computing users corresponding power savings short delays power consumption relatively high decays acceptable delay becomes longer seen users based mobile devices power model prefer local processing tolerate considerable delay using short delays power saving obtained long delays processing delay requirement relaxed power savings offloading diminished local processing power drops exponentially processing delay due power model resulting almost users processing locally users offloading decision relies power model also depends users decisions due interference neighboring cells consequently number active users network also crucial address issue users exist network higher interference created means lower sinr less data rate consequently experienced delay users result percentage local users necessarily absolute number increases increasing number users users cell proposed algorithms could still achieve power saving comparison local computing base line power consumption local computing number users cell fig aggregate power consumption different number users local user percentage power saving percentage number users cell fig power saving percentage local computing users different number users investigate offloading region normal cell edge users find offloading could save power normal users one see large bit stream size low acceptable delay yellow region figures help mobile devices save power fixed delay enlarging bit stream size enter offloading region moreover wider region offloading decision made solving optimization problem simulation results also reveal cell edge users poor channel gain sinr make benefit offloading cloud bad channel condition users need power local processing send data cloud acceptable rate meet delay requirements providing users better sinr using joint transmission might helpful cloud computing local computing maximum acceptable delay maximum acceptable delay local computing cloud computing bit stream size bit stream size offloading regions normal user offloading regions normal user fig offloading regions onclusion paper aggregate power consumption mobile devices crucial aspect mobile cloud computing networks considered accordingly optimization problem aimed minimizing aggregate power users formulated take account practical considerations maximum allowable interference level maximum tolerable delay users considered knowing inherent primary problem applied approximation transform problem convex one proposed two algorithms called solve problem polynomial time algorithm better terms power saving cost complexity therefore suitable used mobile terminal bss high processing resources advantage running users side cost losing optimality simulations demonstrated exist offloading region edge users benefit offloading data cloud finally confirmed results significant enhancement terms power consumption mobile devices could achieved using proposed algorithms eferences masoudi khamidehi cavdar green cloud computing multi cell networks ieee wireless communications networking conference wcnc ieee wcnc san francisco usa mar kumar cloud computing mobile users offloading computation save energy computer vol kwak kim lee chong dream dynamic resource task allocation energy minimization mobile cloud systems ieee journal selected areas communications vol kaewpuang niyato wang hossain framework cooperative resource management mobile cloud computing ieee journal selected areas communications vol zhang niyato wang offloading mobile cloudlet systems intermittent connectivity ieee transactions mobile computing vol zhang wen guan kilper luo mobile cloud computing stochastic wireless channel ieee transactions wireless communications vol claypool claypool latency player actions online games communications acm vol dusi napolitano niccolini longo closer look connections statistical application identification qoe detection ieee communications magazine vol mao survey mobile cloud computing rich media ieee wireless vol piunti cavdar morosi teka del zander energy efficient adaptive cellular network configuration qos guarantee ieee icc mach becvar mobile edge computing survey architecture computation offloading arxiv mao zhang huang letaief mobile edge computing survey research outlook corr vol online available http dou zhang chen enreal resource allocation method scientific workflow executions cloud environment ieee transactions cloud computing vol april kosta aucinas hui mortier zhang thinkair dynamic resource allocation parallel execution cloud mobile code offloading infocom proceedings ieee ieee tout talhi kara mourad smart mobile computation offloading centralized selective approach expert systems applications ellouze gagnaire haddad mobile application offloading algorithm mobile cloud computing mobile cloud computing services engineering mobilecloud ieee international conference ieee barbera kosta mei stefa offload offload bandwidth energy costs mobile cloud computing infocom proceedings ieee ieee liu chen wireless resource scheduling based backoff mobile cloud computing chen liang dong joint offloading decision resource allocation mobile cloud ieee international conference communications icc ieee mahmoodi subbalakshmi sagar cloud offloading enabled mobile devices ieee international conference communications icc ieee huang wang niyato dynamic offloading algorithm mobile computing ieee transactions wireless communications vol sen chiang amuse empowering users offloading throughputdelay tradeoffs ieee transactions mobile computing vol chen jiao efficient computation offloading cloud computing transactions networking vol chen decentralized computation offloading game mobile cloud computing ieee transactions parallel distributed systems vol yang cao yuan han chan framework partitioning execution data stream applications mobile cloud computing acm sigmetrics performance evaluation review vol liu mao zhang letaief computation task scheduling computing systems ieee international symposium information theory isit july cheng shi bai chen computation offloading based mobile cloud computing system ieee international conference communications icc ieee wang sheng wang wang computing partial computation offloading using dynamic voltage scaling satyanarayanan bahl caceres davies case cloudlets mobile computing ieee pervasive computing vol mao zhang letaief joint task offloading scheduling transmit power allocation computing systems corr vol online available http liu lee effective dynamic programming offloading algorithm mobile cloud computing system wireless communications networking conference wcnc ieee ieee meskar todd zhao karakostas energy aware offloading competing users shared communication channel ieee transactions mobile computing vol jan simeone bagheri scutari joint optimization mobile cloud computing user scheduling ieee transactions signal information processing networks vol access physical layer aspects tech masoudi zaefarani mohammadi cavdar energy efficient resource allocation ofdma networks qos guarantees wireless networks kha tuan nguyen fast global optimal power allocation wireless networks local programming ieee transactions wireless communications vol che tuan nguyen joint optimization cooperative beamforming relay assignment wireless relay networks ieee transactions wireless communications vol boyd vandenberghe convex optimization cambridge university press
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preliminary notes termination reasoning sep ton chanh department computer science national university singapore chanhle abstract preliminary note illustrate ideas automated mechanisms termination reasoning termination reasoning logical abduction introduction program termination reasoning gained enormous interest last decade termination reasoning beside traditional approach ranking functions various new approaches proposed principle functional programs polyranking principle programs disjunctive wellfoundedness principle imperative programs reasoning gupta propose approach find recurrent set loop initial configuration infinite executions loop however representative works termination analyses usually considered separate mechanisms words cooperation termination analyses works inspired recent success cooperation safety termination provers finding supporting invariants validity termination arguments strongly believe cooperation termination provers gain success termination analysis benefit analysis excluding behaviors programs example idea excluding program behaviors important termination analyzers based counterexample termination construct termination arguments like armc tools might terminate dealing loops number generated counterexamples infinite past works take advantage analysis reasoning program termination supporting invariants termination arguments synthesized complement unreachable conditions relying existing prover safety prover authors determine domains weakest precondition decidable precondition termination another related approach divides transition relation two subrelations whose termination proved whose behaviors still unknown fixpoint computation program proved terminating inputs set unknown behaviors empty otherwise sufficient condition termination returned opposed approaches aim solve conditional termination problem also conditional problem construct complete picture terminating behaviors program general proposed mechanism incrementally partition unknown program behaviors terminating unknown analysis facilitated safety preconditions program provided users automated safety provers otherwise analysis start trivial precondition true specifically initially prove program showing exit points unreachable given preconditions program might terminate next ranking function synthesizer invoked find termination measure program possibility excluded complete method linear ranking function synthesis fails two remaining possibilities could happen program terminates linear ranking function exist supporting invariant termination proving missing possibilities perform case analysis program preconditions auxiliary condition called potential condition negation intuitively condition ensures program terminate loop condition violated current execution inferred logical abduction seek actual precondition like still possible program terminates without linear ranking function inferred condition actual precondition proved next step analysis otherwise continuously refined steps cases negation potential condition also facilitate termination reasoning strengthens precondition program exit points reachable examples let consider two resembling examples different termination behaviors corresponding loop transition relations fig analyze termination loops interested cases loops conditions satisfied loops thus execute least one time examples initially prove showing exit points unreachable done unsatisfiability check formula transition relation fig examples numerical programs different termination behaviors example terminating input example always nonterminating input condition loop respectively instance exit point loop fig might reachable loop body following formula satisfiable means program possibly terminating check whether loop terminates immediately current execution validity entailment entailment invalid formula satisfiable result shows precondition satisfying loop condition loop might execute one steps know need ranking function synthesis tool find witness program termination two simple checks avoid invoke synthesis tool early trivial cases termination considered loop however receive failure result synthesis tool consequently linear ranking function loop supporting invariant helps separate terminating execution loop missing termination proving program find missing invariant firstly infer potential condition ensures loop condition violated current execution opposed condition inferred finding potential ranking function condition logically inferred entailment thus handle loop conditions general structure disjunctions trivial solution potential condition loop condition thus negation condition sufficient condition program termination loop terminates current execution however need stronger condition obtained via logical abduction abductive constraint validity entailment usually inferred quantifier elimination universal formula set variables determined minimum satisfying assignment msa defined users another approach suitable abductive inference constraint solving approach unknown constraint would inferred farkas lemma solver contrast designed lightweight abductive inference linear arithmetic balance number variables generality inferred constraints approach require expensive quantifier elimination calculation msa constraint solver back example inferred potential condition perform case analysis condition partitioning transition relation opposed approach syntacticly partition transition relation based condition decrease potential ranking function boundedness function respectively applying procedure prove execution corresponding relation always terminates termination proof successful auxiliary condition also proved supporting invariant execution corresponding relation also terminates eventually reaches terminating case thus program always terminate input example similarly infer potential condition condition also actual condition loop case mark possibly terminating case execution reach either base case nonterminating case according initial reachability checks another advantage approach termination loop analyzed modular fashwhile ion supporting invariants termination proof discovered loop rather initial contexts might complicated analyze let fig modified consider modified version example fig version initial value determined compliple cated function consequently supporting invariant complicated ranking function hard discover due complexity analysis example tion analyze loop isolation show loop terminates input example fig thus conclude whole program terminates assumption function also terminates references bozga iosif deciding conditional termination proc tacas pages bradley manna sipma linear ranking reachability proc cav pages bradley manna sipma polyranking principle proc icalp pages brockschmidt cook fuhs better termination proving cooperation proc cav pages cook gulwani rybalchenko sagiv proving conditional termination proc cav pages cook podelski rybalchenko termination proofs systems code proc pldi pages dillig dillig explain tool performing abductive inference proc cav pages ganty genaim proving termination starting end proc cav pages gupta henzinger majumdar rybalchenko proving proc popl pages harris lal nori rajamani alternation termination proc sas pages larraz oliveras rubio proving termination imperative programs using proc fmcad lee jones principle program termination proc popl pages peirce collected papers charles sanders peirce volume harvard university press podelski rybalchenko complete method synthesis linear ranking functions proc vmcai pages podelski rybalchenko armc logical choice software model checking abstraction refinement practical aspects declarative languages pages
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control scheduling complex dynamical networks erfan fabio jorge nov department mechanical aerospace engineering university california san diego department mechanical engineering university california riverside corresponding author email enozari abstract despite extensive research remarkable advancements control complex networks control schedules tics still dominate literature due simplicity fact potential benefits control schedules tvcs remained largely uncharacterized yet tvcs potential significantly enhance network controllability tics especially applied large networks paper study networks linear dynamics analyze role network structure tvcs analysis new notion nodal communicability show optimal tvcs involves actuation central nodes appropriate spatial scales times consequently show network determine whether extent tvcs outperforms conventional policies based tics several analytical results numerical examples support illustrate relationship many natural systems ranging nervous system power transportation grids societies exhibit dynamic behaviors evolve sparse complex network ability control network dynamics theoretically challenging problem also barrier fundamental breakthroughs across science engineering multiple studies addressed various aspects problem several fundamental questions remain unanswered including extent capability controlling different set nodes time improve controllability complex networked systems controllability dynamical network network supports temporal evolution set nodal states classically defined possibility steering state arbitrarily around state space application external inputs actuation certain control nodes raises fundamental question choice control nodes affect network controllability hereafter refer control scheduling problem notice classical setting attention paid possibility arbitrarily steering network state difficulty energy cost motivated introduction several controllability metrics quantify required effort control scheduling problem comprehensive solution remained elusive works collectively revealed role several factors control scheduling problem network size structure nodal dynamics centralities number control nodes choice controllability metric majority literature however implicitly relies assumption tics namely control node fixed time depending specific network structure assumption may come expense significant limitation controllability especially systems distant nodes inevitably exist relative control point intuitively possibility tvcs namely ability control different nodes different times allows targeted interventions different network locations ultimately decrease control effort accomplish desired task hand practical standpoint implementation tvcs requires ability geographically relocate actuators presence actuation mechanisms different ideally network nodes sophisticated control policies leads critical benefits tvcs implementation costs received enough attention literature significant potential schedules control also sensing dual interpretation control led design sub optimal sensor control scheduling algorithms recent years constituting notable leap forward benchmark methods developed paper works oblivious fundamental question whether extent tvcs provides improvement network controllability compared tics previous work studied former question whether tvcs provides improvement tics case undirected networks consider general controllability metrics importantly addressed latter question large relative improvement network controllability given benefits costs tvcs clear answer question vital practical application tvcs complex networks paper address two questions context directed linear network dynamics show new notion nodal centrality define plays fundamental role tvcs notion measures centrality node network different spatial scales based distinction local global nodal centralities show optimal control point every time instance node largest centrality appropriate scale node largest appropriate accordingly main conclusion benefit tvcs directly related scaleheterogeneity central nodes network benefit gained networks highest centrality attained various nodes different spatial scales benefit starts decay fewer nodes dominate network scales provide analytical results extensive case studies synthetic real networks support illustrate conclusion results model time according control input input vector time time horizon simplicity exposition consider one control input time discussion generalizable networks supplementary note index node control signal applied time equal column identity matrix purpose applying control input steer network state initial state desired state possible called controllable case controllability gramian invertible case minimal energy required steering network origin equals xft methods xft denote transpose vector inverse matrix respectively therefore reachable set namely set xft direction length axes given eigenvectors square root eigenvalues respectively methods intuitively larger reachability controllable network dynamics equation quantify large several measures based eigenvalues proposed literature including trace denoted measuring average squared length axes trace denoted measuring average control energy random target states smallest eigenvalue denoted measuring controllability determinant denoted det measuring volume metric benefits limitations elaborate following section consider network nodes communicate discrete time weighted general directed graph denote set nodes edges respectively adjacency matrix aij denotes weight edge node node see supplementary note methods obtaining network connectivity structure node state value evolves time interaction node neighbors assume node linear dynamics one node similar expression holds arbitrary customary controlled exogenously time therefore evaluate control energy starting network unforced equiliboverall network state evolves rium scheduling control measures rely heavily assume aforementioned controllability measures seek choose control nodes equivalently optimally conventional approach literature problem reduce search space constant choices control node choose maximized namely control scheduling tics problem formally specified max tion trades average controllability directions state space despite significant increase size complexity core principles outlined apply controllability networks large size networks however imposes new constraints choice controllability measure make use det numerically infeasible theoretically methods result resort particular choice controllability measure networks beyond since measure smallest sensitivity choice figure main advantage tics simplicity thewe expect network benefits tvcs using oretical computational implementation perspectives choice equation also benefit using however simplicity comes possibly significant cost measures converse necesin terms network controllability compared case sarily true networks significantly benewhere control nodes independently chofit tvcs using measures show benefit sen namely terms figure illustrates air transmax portation network among busiest airports united states comprising nodes size approach namely control scheduling network makes use tvcs least good tics potential det infeasible using choice equation improve network controllability figure illustrates see improvement controllability small network nodes together optimal network verifying expectation benefits values equations relative advantage tvcs tvcs tics defined spite potential benefit tvcs usually higher computational implementation costs need max max fmax taken account include higher computathree observations worth highlighting first value tional cost computing optimal tvcs extremely dependent choice controllability installing actuator several ideally nodes measure different choices lead orders network networks benefit tvcs tude change second relative advantage tvcs alike simple directed chain network size tics significant choices figure gains absolutely benefit ity measure minimum improvement tvcs independently choice controllability meafor choice finally even sure figure similarly values also mal tvcs orders magnitude less observed larger complex networks indicating indicating inevitable existence optimal tvcs involves control single node rections state note except times figure observations collectively raise fundamental fact results smallest value relation networks networks benefit tive measures consistently observed synthetic realworld networks stems fact smallest tvcs one distinguish ease reference call networks class sensitivity greatest robustness choice control schedule controllability improves respectively words network belongs class larger number nodes simultaneously controlled however class otherwise followthis examples shows efficient controllability maintained directions state space even using tvcs small ing restrict attention choice controllability measure equation due applicability netnetworks control nodes work sizes carry thorough analysis properties ease reference let denote index order address aforementioned fundamental question node largest value formally arg maxi according discussion theory dynamical networks relation dynamic networks introduce new notion nodal communicability plays pivotal role distinguishing class class networks shown solution tvcs equation consists indices largest diagonal entries methods words optimal control node time precisely index largest diagonal entry motivates definition node shown results rankings nodes left eigenvector centrality squared thus taken definition methods supplementary note figure illustrates evolution function sample network nodes perhaps salient property extent relies local interactions among nodes recall entry equals total number paths length node graph weighted path counts weight equal product weights edges equation see equals sum squares total weighted number paths length ending node words depends connections node outneighbors independent rest network therefore local notion centrality small incorporates global information grows particular closely related centrality node left eigenvector centrality node methods supplementary note scaling property illustrated figure example network nodes core connection tvcs see optimal tvcs involves application node highest global centrality gradually moving control point apply node highest local intuition behind procedure simple control input enough time propagate network highest node controlled reach control horizon control input time steps disseminate network hence optimality nodes role tvcs immediate implication distinguishing class networks network belongs class nodes highest distinct supplementary note consider networks figure color intensity node indicates value size corresponds value clearly first largest darkest nodes distinct figure close correlation nodal size darkness figure illustrating root cause difference benefiting tvcs network still possible network belongs class fact half networks still belong figure however networks value average turn value quickly decreases dominance node rest network nodes figure strong indication practical purposes test based valid indicator whether network benefits tvcs furthermore case undirected networks possible analytically prove network belongs class certain conditions based adjacency matrix satisfied shown supplementary note conditions include distinguishing class net networks eigenvector centrality one node significantly higher rest network works based notice control node time arbitrary next use scaling property unveil network structures benefit tvcs fmax fmax det fmax fmax det figure advantage tvcs dynamic networks small example network nodes thickness edge illustrates weight aij optimal values tics tvcs equations respectively relative tvcs advantage equation network air transportation network among busiest airports united states see table details network undirected dynamical adjacency matrix computed static connectivity using transmission method supplementary note example network significantly benefits tvcs small example network size benefit tvcs optimal values tics tvcs equations respectively relative tvcs advantage equation network see network benefit tvcs independently choice controllability metric social network students university california irvine see uci forum table details similar network undirected adjacency matrix computed using transmission method supplementary note network however benefit tvcs controllability measure equation used due large size network cases color intensity size nodes represent values respectively close correlation nodal size color intensity darkest nodes also largest case root cause difference figure dynamical networks example network nodes illustration dependence nodal thickness edges proportional weights evolution functions although functions originally defined integer values extended domain real numbers better illustration crossings oscillatory behavior oscillatory behavior arises eigenvalues otherwise strictly convex example network nodes illustration scaling property node whose computed node depicted red node depicted red determined incoming paths see depends immediate neighbors grows encodes global information fair concern however exists regarding minimum network dynamics dominated largest eigenvalue totally disconnected network nonzero size manipulation needed make tvcs allmanifest excessively high prescribed approach extreme case networks may infeasible practice nevertheless among net networks eigenvector centrality works various size structure random manipulations nodes determined weight link norm norm average central one canonical example star network sufficient figure see largest manipuno small lations needed manifest subnetworks iii networks three distinct eigenvalues total size network complete bipartite networks connected strongly size manifest subnetwork extremely small manipular networks cones strongly regular graphs ulations focused small subset nodes thus star networks also canonical efficient extremely large manifest subnetworks majority nodes accessible control example little restriction tvcs general abstraction cases network finally figure also shows comparison terms belongs class contains sufficiently distinct central controllability approach node reinforces main conclusion alternative approach selecting optimal tvcs class networks multiple additional constraint control nodes must mantral inclusion relationships various ifest without manipulation dynamics classes networks introduced section results tvcs comrized figure parison fair normalize network spectral radius largest magnitude eigenvalues effects network manipulation compare optimal value tvcs equation see amount relative advantage produced timal control scheduling manifest subnetwork manipulation comparable many applications control scheduling relative size manipulation except nodes control manifest subnetworks nodes macase manifest subset nodes available nipulation advantage two times size application control inputs remaining nodes latent controller natural solution would tvcs synthetic real networks choose control nodes optimally among manifest nodes another solution however discuss benefits tvcs relation late dynamics among manifest nodes network structure several examples synthetic real optimal control nodes computed without networks start classical deterministic examtions control scheduling lie among manifest nodes ples undirected line ring star networks figure due simple structure time provided manipulation possible analytically guarantee latter scheme networks analytically computed closed form manipulating dynamics always supplementary note using results follows enforce entirely composed line star networks optimal control ifest nodes optimal tvcs provided node always center node two center tion sufficiently strong acyclic supplementary nodes line even number nodes optinote requirements clear interpretations mal control node arbitrary ring network notice first depending large size manifest cases homogeneity networks subnetwork central nodes already results single node greatest centrality manipulation larger manipulation may necessary scales see supplementary note examples nonturn central nodes various scales homogeneous star networks second manifest nodes central nodes thus belong class next analyze role tvcs three classes become central arbitrarily global scales manipulation must contain paths probabilistic complex networks widely used capbitrarily long lengths absent acyclic networks ture behavior various dynamical networks dynamical networks thm thm benefit tvcs figure role distinguishing networks class proportion networks randomly generated networks networks belong class number drops less among networks networks node greatest local global centralities average networks function dominance node vertical bars represent one standard deviation intervals node dominance rest network measure distinctly larger respectively methods random connectivity matrices generated uniform sparsity uniform pairwise connectivity transformed adjacency matrices using transmission method supplementary note venn diagram illustrating decomposition dynamical networks based extent benefit tvcs color gradient depiction extent measured equation darker areas correspond higher shown class networks subset provides good approximation include random networks networks networks network characteristic properties properties lead different behaviors tvcs average networks computed various values network parameters figure networks general small decays networks especially large extremely homogeneous homogeneity increased transmission method leading network matrix extremely insensitive choice control nodes connectivity structure networks contrast extremely inhomogeneous one sometimes highly central nodes hierarchy peripheral leafs one would expect implies small since center node highest centrality scales supplementary figure however connectivity matrix transformed using transmission method incoming links nodes made uniform adding turns make centrality levels nodes comparable leading high observed notice underlying connectivity structures still highly inhomogeneous distinguishing mogeneous networks notice growth rate increased smaller networks tend towards complete graphs high values shift larger last class probabilistic networks networks broadest range values significant one would expect low near corresponding regular ring lattice networks respectively sufficiently high probability multiple nodes close many rewired links increasing centrality yet low probability nodes nodes close rewired alike resulting heterogeneous central nodes high heterogeneity increased larger networks possibilities rewiring every edge finally used tools concepts introduced far analyze tvcs several dynamical networks table networks chosen wide range application domains neuronal networks transportation social networks according type dynamics evolving network used either transmission induction method obtain dynamical adjacency matrix static connectivity supplementary note figure simple networks line network ring network star network networks undirected homogeneous edge weights networks analytically computed supplementary note concluding networks belong class optimal control node depicted red case optimal control node arbitrary ring network due symmetry figure manipulation manifest subnetworks order obtain optimal tvcs horizontal axis represents percentage manifest nodes network red show minimum size manipulation needed optimal tvcs include manifest nodes relative size initial adjacency matrix measured induced matrix blue depict optimal maximal value case minimal manifest manipulation applied relative maximal value subject constraint control nodes manifest former manipulation without constraints control nodes latter manipulation control node constraints results random networks sizes otherwise similar figure markers represent average values bars represent one standard deviation intervals cases overall adjacency matrix normalized spectral radius fairness comparison see manifest subnetworks hardest yet fruitful manipulate computed network using variable time horizon results ranging different networks large variations even within category signifies potential benefits tvcs possibility redundancy contrast pivotal discussion last column also indicated whether local global central nodes coincide network recall sufficient necessary condition network class figure supplementary note though sufficient simple metric correctly classify class members among networks except westernus power network marginally holds dominance figure discussion despite breadth depth existing literature controllability complex networks control scheduling significant potential tvcs greatly overlooked work strives explore advantages tvcs linear dynamical networks obtaining theoretical computational relationships advantages network structure using measures controllability showed tvcs significantly enhance controllability many synthetic real networks motivated pursuit identifying properties based network structure explain much tvcs beneficial using newly introduced notion table characteristics networks studied paper directed dominance ref bctnet fmri cocomac bctnet cat elegans transportation airus airglobal chicago gene regulatory coli ppi yeast stelzl figeys vidal power westernus food florida lrl social facebook group southern women uci uci forum freeman eies dolphins trust physicians org consult advice org consult value org advice org aware category name neuronal network reported number nodes number edges bidirectional edge counted twice whether network directed method used obtaining dynamical adjacency matrix static connectivity value equation whether local global central nodes coincide since value function chosen value largest network detailed descriptions datasets provided supplementary note figure average probabilistic networks horizontal axis determines size network cases vertical axis determines values corresponding parameters network edge probability growth link attachment rate rewiring probability constructing unweighted connectivity according algorithm standard uniformly random weights assigned edge converted using transmission method supplementary note value network parameter coarse mesh points networks generated average computed smoothly interpolated fine mesh matlab csaps communicability showed central nodes network main cause correlate tvcs advantages network several distinct central nodes different scales optimal tvcs involves starting control global central nodes gradually moving towards local ones time horizon approached hand single node acquires highest centrality scales optimal tvcs prescribes sole control node entire horizon leading optimality tics striking finding defied expectations effect network dynamics beyond raw connectivity structure tvcs differentiated raw connectivity structure network obtained using specific field knowledge measure relative strength nodal connections dynamical adjacency matrix determines evolution network state time depending nature network state proposed two methods transmission induction obtaining dynamical adjacency matrix static connectivity effects methods however noteworthy benefits tvcs even though underlying network connectivity table supplementary figure transmission method significantly enhances merit tvcs induction method depresses compared raw connectivity believe reason former additional homogeneity transmission method introduces among nodes latter due conversion continuous dynamics enables connections even small sampling times due fact interactions occur infinitesimal intervals continuous time results suggest controllability network dynamics function structural connectivity also greatly relies type dynamics evolving network aspect received little attention existing literature warrants future research focus discussion far single input networks one node controlled time order enhance simplicity clarity concepts nevertheless results straightforward generalizations networks supplementary note denotes number control inputs optimal tvcs involves applying control inputs nodes highest centralities appropriate scale every time instance nodes largest controlled every time instance clear additional flexibility due additional inputs makes larger networks nevertheless additional flexibility also makes tics significantly efficient therefore immediately clear whether enlargement also entails larger networks size sparsity fact increasing reduces average classes networks supplementary figure suggesting additional flexibility advantageous tics tvcs regardless number inputs portant implicit assumption work number limited nodes controlled every time instance may first seem since tvcs requires essence installation actuators many nodes network therefore one might wonder limit control nodes every time instance nodes ready actuation answer lies within practical limitations actuators ideal actuators distributing control energy many nodes possible indeed optimal however possible many scenarios including actuators exhibit nonlinear deadzone behaviors one requires sizable activation energy actuators controlled via communication channels limited capacity small number devices simultaneously operated iii actuators geographically disperse precise coordination becomes difficult simultaneous control proximal nodes results actuator interference although dynamics real networks degrees nonlinearity analysis linear ized dynamics standard first step analysis dynamical properties complex networks mainly due fact stability controllability linearized dynamics nonlinear network implies properties locally original nonlinear dynamics making linear dynamics powerful tool analyzing many dynamical properties general intractable nonlinear dynamics local validity linearization however main limitation work particularly networks change state significant relative size domain linearization valid generalization work linear dynamics namely instead equation warranted next step future exploration role tvcs general nonlinear networks methods optimal control linear networks reachability ellipsoid assume linear network dynamics equation controllable time invertible equivalently positive definite usually infinitely many choices take network one smallest energy given thm immediate verify gives minimal energy therefore ability set given xft since positive definite axes aligned eigenvectors let cvi xft showing axis lengths given square roots eigenvalues measures controllability measures common measures network controllability due relationship gramian eigenvalues minimal control energy outlined measures quantify large eigenvalues different ways let denote eigenvalues widely used measures det clear relationships measures except approach property behavior measure critical difference three measures rest discussion let det since network controllable guarantees network controllability major disadvantage small networks controllability directions state space achievable desirable size network grows however typically decays exponentially fast zero irrespective network structure exponential decay controllability even evident example network figure comprising nodes computationally means turn quickly drops machine precision grows fact arithmetics happens making tvcs equation numerically infeasible involves comparison different zero machine accuracy notice computational complexity tvcs grows due tvcs enforcing use greedy algorithms even machine precision concern see references therein details addition computational aspects tvcs exponential decay also theoretical implications choice using tvcs seeks assign control nodes controllability maintained directions state space special emphasis directions use hand involves maximizing average gramian eigenvalues usually strengthens largest eigenvalues spares smallest ones large networks latter general realistic controllability hardly needed directions state space discussed detail seems case structural brain networks paper shows maximized controlling specific brain regions long identified structural core hubs cerebral cortex gramian close singular finally due strong dependence often observe significantly less sensitive choice control nodes leading orders magnitude smaller figure means small subclass networks benefit tvcs measured also clear interpretation since maintaining controllability directions state space requires broader distribution control nodes facilitates reach control action nodes network optimal tvcs tics consider tvcs problem equation using definition controllability gramian equation invariance property trace cyclic permutations write therefore max max see supplementary note definitions various eigendecomposition based notions network centrality recall centrality node defined respectively diout aji therefore network unweighted edges weight diout centrality result ranking nodes particular node largest edge weights become heterogenous two rankings become less correlated putting emphasis stronger weights similar relation exists left eigenvector centrality show next let right left eigenvectors respectively normalized since network assumption strongly connected aperiodic lim thus lim lim given dividing change ranking nodes define since squaring numbers preserves order nodal rankings based left eigenvector centrality identical term diagonal entry therefore optimization equation boils finding largest diagonal element applying node hand tics problem equation one instead find largest diagonal entry apply controls inputs node clearly respect tvcs degree eigenvector centrality notion introduced article close connections degree eigenvector centrality limit cases respectively nodal dominance among networks nodes greatest coincide higher chance general network belongs class however half networks still belong class meaning exists assess importance optimal control nodes define dominance node rest network follows let index node second largest largest removing similarly let index second largest define dominance min small dominance indicates another node similar value large dominance indication large gap next largest references kalman mathematical description linear dynamical systems journal society industrial applied mathematics series control vol liu slotine controllability complex networks nature vol cowan chastain vilhena freudenberg bergstrom nodal dynamics degree distributions determine structural controllability complex networks plos one vol olshevsky minimal controllability problems ieee transactions control network systems vol yan ren lai lai controlling complex networks much energy needed physical review letters vol nozari pasqualetti versus actuator scheduling complex networks american control conference seattle may chen linear system theory design new york usa oxford university press weber analysis optimization certain qualities controllability observability linear dynamical systems automatica vol godsil royle algebraic graph theory ser graduate texts mathematics springer vol rubinov sporns complex network measures brain connectivity uses interpretations neuroimage vol bakker wachtler diesmann cocomac future databases frontiers neuroinformatics vol watts strogatz collective dynamics networks nature vol pasqualetti zampieri bullo controllability metrics limitations algorithms complex networks ieee transactions control network systems vol colizza vespignani processes metapopulation models heterogeneous networks nature physics vol summers lygeros optimal sensor actuator placement complex dynamical networks ifac world congress cape town south africa opsahl anchorage important binary ties sample selection available http summers cortesi lygeros submodularity controllability complex dynamical networks ieee transactions control network systems vol tzoumas rahimian pappas jadbabaie minimal actuator placement bounds control effort ieee transactions control network systems vol zhao zhang abate tomlin optimal solutions linear sensor scheduling problem ieee transactions automatic control vol jawaid smith submodularity greedy algorithms sensor scheduling linear dynamical systems automatica vol zhao pasqualetti scheduling control nodes improved network controllability ieee conf decision control las vegas han zhang shi optimal sensor scheduling multiple linear dynamical systems automatica vol eash chon lee boyce equilibrium traffic assignment aggregated highway network sketch planning transportation research record vol boyce chon ferris lee lin eash implementation evaluation combined models urban travel location sketch planning network chicago area transportation study kim shim shin lee ecolinet database cofunctional gene network escherichia coli database vol zhao cai xue zhu zhang sun ling zhang chen topological structure analysis interaction network budding yeast nucleic acids research vol stelzl worm lalowski haenig brembeck goehler stroedicke zenkner schoenherr koeppen timm mintzlaff abraham bock kietzmann goedde toksz droege krobitsch korn birchmeier lehrach wanker human interaction network resource annotating proteome cell vol ewing chu elisma taylor climie robinson connor taylor dharsee heilbut moore zhang ornatsky bukhman ethier sheng vasilescu lambert duewel stewart kuehl hogue colwill gladwish muskat kinach adams moran morin topaloglou figeys mapping human interactions mass spectrometry molecular systems biology vol rual venkatesan hao dricot berriz gibbons dreze towards map human interaction network nature ulanowicz heymans egnotovich network analysis trophic dynamics south florida ecosystems graminoid ecosystem annual report united states geological service biological resources division ref umces cbl chesapeake biological laboratory university maryland martinez magnuson kratz sierszen artifacts attributes effects resolution little rock lake food web ecological monographs vol leskovec mcauley learning discover social circles ego networks advances neural information processing systems nips yin benson leskovec gleich local graph clustering proceedings acm sigkdd international conference knowledge discovery data mining acm leskovec kleinberg faloutsos graph evolution densification shrinking diameters acm trans knowl discov data vol davis gardner gardner deep south chicago university chicago press opsahl panzarasa clustering weighted networks social networks vol opsahl triadic closure networks redefining global local clustering coefficients social networks vol freeman freeman networkers network study impact new communications medium sociometric structure ser social sciences research reports school social sciences university lusseau schneider boisseau haase slooten dawson bottlenose dolphin community doubtful sound features large proportion associations behavioral ecology sociobiology vol coleman katz menzel diffusion innovation among physicians sociometry cross parker hidden power social networks understanding work really gets done organizations harvard business school press pasqualetti cieslak telesford kahn medaglia vettel miller grafton bassett controllability structural brain networks nature communications vol author contributions authors designed research wrote paper gathered analyzed data networks additional information competing financial interests authors declare competing financial interests
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jun complete reducibility question conjugacy classes example tomohiro uchiyama national center theoretical sciences mathematics division sec roosevelt national taiwan university taipei taiwan email june abstract let nonperfect separably closed field let connected reductive algebraic group defined study rationality problems serre notion complete reducibility subgroups particular present new example subgroup type characteristic reducible gcompletely reducible vice versa new known examples exceptional type also find new counterexample question representations finite groups type problem concerning number conjugacy classes also considered notion nonseparable subgroups plays crucial role constructions keywords algebraic groups complete reducibility rationality geometric invariant theory representations finite groups conjugacy classes introduction let field let algebraic closure let connected affine algebraic regard algebraic group together choice sense borel say reductive unipotent radical trivial throughout always connected reductive paper continue study rationality problems complete reducibility subgroups subgroup mean possibly closed subgroup following serre sec definition subgroup called reducible short whenever contained parabolic subgroup contained levi subgroup particular contained proper parabolic subgroup called short far studies complete reducibility complete reducibility see example say subgroup much known complete reducibility especially nonperfect except theoretical results important examples see sec thm thm thm sec bate found several examples vice versa examples exceptional type constructions intricate first main result paper following theorem let nonperfect separably closed field characteristic let simple type exists vice versa comments order first one embed inside levi subgroup since subgroup contained subgroup proposition one might argue new example really new however checked example different example thm thm thm first example classical second nonperfectness essential theorem view following thm proposition let subgroup separable closure particular perfect subgroup proposition deep depends recently proved center conjecture tits see conjecture spherical buildings third theorem important actually difficult find desired property construction subgroup essential nonseparable write lie lie algebra recall def definition subgroup nonseparable dimension lie strictly smaller dimension acts via adjoint action words centralizer sense def smooth exhibit importance nonseparability proof theorem proper nonseparable hard find handful examples known sec thm thm thm known good every subgroup separable thm thus find nonseparable subgroup forced work small see separability rest section assume algebraically closed asked question let triple reductive groups connected vice versa general answer either direction easy find counterexample reverse direction take sits inside via adjoint representation counterexamples see counterexample forward direction hard find handful examples known thm thm sec examples exceptional type second main result theorem let characteristic let simple type exists pair reductive subgroups reductive pair recall pair reductive groups called reductive pair liem direct summand see reductive pairs construction nonseparablity essential thm proposition suppose reductive pair let subgroup separable move problem slightly different flavor let finite group representation reductive group mean homomorphism write hom set representations group acts hom conjugation let sylow sec asked question let reductive algebraic group defined algebraically closed field characteristic let hom finitely many representations hom known general answer two counterexamples known one type type thm third main result paper theorem let characteristic let simple type exists finite group sylow representations hom conjugate restrictions pairwise conjugate note nonseparability plays crucial role proof theorem paper reader see seemingly unrelated questions rationality problems reducibility problem conjugacy classes related main results concerning problems theorems based mechanism nonseparability plus modifications however completely clear yet least author exactly problems related main purpose paper give chance reader look problems relatively easy example type stimulate research relations problems finally consider problem number conjugacy classes given let act simultaneous conjugation ggn slodowy proved following result applying richardson beautiful tangent space argument sec lem proposition let reductive subgroup reductive algebraic group defined algebraically closed field let let let subgroup generated suppose reductive pair separable intersection finite union classes proposition many consequences see sec example main result conjugacy classes theorem let characteristic let simple type let subsystem subgroup type exists tuple infinite union classes structure paper section set notation show preliminary results section prove first main result theorem concerning rationality problem complete reducibility section prove rationality result theorem related center conjecture section give short proof second main result complete reducibility theorem using recent result geometric invariant theory proposition section prove theorem giving new counterexample question finally section consider problem conjugacy classes prove theorem preliminaries throughout denote separably closed field references algebraic groups let possibly affine algebraic group write identity component write derived group reductive group called simple algebraic group connected proper normal subgroups finite write set respectively simply say characters cocharacters fix maximal exists cor splits since separably closed let denote set roots respect sometimes write let write corresponding root subgroup define let let coroot corresponding let denote reflection corresponding weyl group acts set roots following formula lem prop lem choose next result prop shows complete reducibility behaves nicely central isogenies paper specify isogeny type argument works isogeny type anyway note algebraically closed centrality assumption necessary proposition definition let reductive central ker central ker differential identity lie algebra proposition let reductive let subgroups subgroups respectively let central next result thm used repeatedly reduce problems reducibility reducibility levi subgroup proposition suppose subgroup contained levi subgroup recall characterizations parabolic subgroups levi subgroups unipotent radicals terms cocharacters prop characterizations essential translate results complete reducibility language git see example definition let affine let affine say lim exists exists necessarily unique whose restriction limit exists set lim definition let define lim exists lim lim parabolic subgroup levi subgroup unipotent radical sec parabolic subgroups levi subgroups arise way since separably closed well known note levi subgroups parabolic subgroup lem iii let reductive natural inclusion groups let write parabolic subgroup corresponding parabolic subgroup corresponding clear recall following geometric characterization complete reducibility via git suppose subgroup generated acts simultaneous conjugation proposition subgroup closed combining proposition recent result git thm proposition let subgroup let suppose exists proof theorem let simple algebraic group type defined nonperfect field characteristic fix maximal borel subgroup let set roots corresponding set positive roots corresponding following dynkin diagram defines set simple roots label following corresponding negative roos defined accordingly note roots correspond respectively let let pick let define first main result section proposition moreover proof first using commutation relations lem prop obtain since commutes clear since generated show sufficient show since since contained proposition enough show inspection easy since next show suppose contrary clearly contained exists levi subgroup containing lem iii exists contained prop set using labelling positive roots compute acts using commutation relations thus must last equation gives impossible since done remark computations see curve contained corresponding element lie contained argument proof prop shows dim strictly smaller dim fact combining thm thm separable move result section use second let let define proposition moreover proof clearly first show note thus see contained contained lemma unique proper parabolic subgroup containing proof suppose proper parabolic subgroup containing proof proposition shown exists levi subgroup containing since contained since levi subgroups lem iii without loss set centralizes recall thm cru cru open set opposite containing lemma proof first equation see contained since also contained contained set using equation commutation relations obtain cru must cru must cru thus conclude cru similarly show cru direct computation shows done since centralizes lemma yields set bruhat decomposition one following forms rule second case suppose second form note enough show since contained assume longest element weyl group using compute acts root subgroup particular thus must first form thus done lemma proof suppose contrary since thm thus put since parabolic subgroups thus kpoint rational version bruhat decomposition thm exists unique unique contradiction since lemmas show lemma proof since using commutation relations see note contains commute element also since commute element thus conclude unipotent classical result prop see since normal subgroup cor tits center conjecture tits conjectured following conjecture let spherical building let convex contractible simplicial subcomplex automorphism group stabilizing exists simplex fixed center conjecture tits proved analyses tits leeb recently uniform proof given relation theory complete reducibility serre showed proposition let reductive let building fixed point subcomplex convex contractible identify set proper subgroups usual sense tits note subgroup induces automorphism group stabilizing thus combining center conjecture proposition obtain proposition subgroup exists proper subgroup containing proposition essential tool prove various theoretical results complete reducibility nonperfect asked following rem question exist proper subgroup containing hcg answer yes perfect since case set points dense since assume result follows proposition main result section present counterexample question nonperfect theorem let nonperfect characteristic let simple type exists contained proper subgroup remark rem mentioned nonperfect characteristic exists unipotent element type contained proper subgroup proof note generates cyclic subgroup recall unipotent element called embedded unipotent radical proper kparabolic subgroup theorem nonabelian version result also assumption necessary proof keep previous section set let clear thus running similar argument proof lemma previous section find proper parabolic subgroup containing since clearly contain therefore proper parabolic subgroup containing thus proper parabolic subgroup containing hcg proof theorem section assume algebraically closed let hypothesis let let let define argument previous section let proposition proof let let natural projection let lim see since centralizes proposition shows question proof theorem let odd let dihedral group order let let sylow let hypothesis choose let define hom computation shows let thus pairwise conjugate suppose conjugate exists since must let note since implies easy computation shows thus since simple group type conjugate done conjugacy classes proof theorem proof let hypothesis let using commutation relations let pick let define argument proof lem obtain cru since standard result exists let let set similar argument proof lem yields infinite union classes crucial thing existence curve tangent tangent words nonseparable let canonical projection since shown previous section prop done acknowledgements research supported postdoctoral fellowship national center theoretical sciences national taiwan university references bate herpel martin rational theorem appear math bate herpel martin spherical buildings pacific bate martin geometric approach complete reducibility invent bate martin complete reducibility separable field extensions acad sci bate martin question representations finite groups reductive groups israel bate martin tange complete reducibility separability trans amer math bate martin tange closed orbits uniform geometric invariant theory trans amer math borel linear algebraic groups springer graduate texts mathematics second enlarged edition borel tits groupes publ math ihes borel tits unipotents paraboliques des groupes invent math carter simple groups lie type john wiley sons conrad gabber prasad groups cambridge second edition goodbourn reductive pairs arising representations algebra herpel smoothness centralizers reductive groups trans amer math humphreys linear algebraic groups springer graduate texts mathematics donovan conjecture crossed products algebraic group actions israel math leeb centre conjecture spherical buildings types geom funct liebeck seitz reductive subgroups exceptional algebraic groups mem amer math tits centre conjecture buildings algebra weiss receding polar regions spherical building center conjecture ann inst fourier center conjecture thick spherical buildings geom dedicata richardson conjugacy classes lie algebras algebraic groups ann richardson orbits algebraic groups lie groups bull austral math soc richardson conjugacy classes lie algebras algebraic groups duke math serre bourbaki vol slodowy two notes finiteness problem representation theory finite groups austral math soc lect ser algebraic groups lie groups cambridge univ press cambridge springer linear algebraic groups progress mathematics second edition stewart reducibility exceptional algebraic groups phd thesis imperial college london stewart reducible subgroups exceptional groups int math res thomas simple irreducible subgroups exceptional algebraic groups algebra tits groupes isotropes colloq des groupes bruxelles tits buildings spherical type finite springer lecture notes mathematics first edition uchiyama complete reducibility subgroups reductive algebraic groups nonperfect fields arxiv appear comm algebra uchiyama complete reducibility examples application question appear group theory uchiyama separability complete reducibility subgroups weyl group simple algebraic group type algebra uchiyama complete reducibility subgroups reductive algebraic groups nonperfect fields algebra vinberg invariants set matrices lie theory
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mar note howson property inverse semigroups peter jones department mathematics statistics computer science marquette university milwaukee usa march abstract algebra howson property intersection two finitely generated subalgebras finitely generated simple necessary sufficient condition given howson property hold inverse semigroup finitely many idempotents addition shown monogenic inverse semigroup howson property keywords howson property monogenic mathematics subject classification introduction algebra howson property intersection two finitely generated subalgebras finitely generated eponym howson stems shown free groups property motivated recent work silva soares find remarkably simple characterization inverse semigroups howson property assumption semilattice idempotents finite howson property holds true maximal subgroups specialized semigroups immediately deduced howson property holds true maximal group image thus generalize main theorem using elementary methods based green relations technical ideas motivated techniques authors work full inverse subsemigroups throughout inverse semigroups regarded unary semigroups thus group howson property regarded group true regarded inverse semigroup perhaps deepest earlier work howson property inverse semigroups trotter author showed although free inverse semigroups rank one howson property property fails free inverse semigroups rank greater one silva also showed however intersection two monogenic inverse subsemigroups free inverse semigroup finitely generated providing technical background connect work cited paper concerns inverse semigroups semidirect products semilattices groups first main result theorem resulting semigroup howson property group proved hypothesis semilattice finite thereby follows cited theorem see corollary although make observation may note fact corollary may decuced result virtue carroll embedding theorem see discussion following cited corollary details preliminaries refer reader text petrich inverse semigroups general including background green relations congruences semilattice idempotents inverse semigroup denoted hxi denotes inverse subsemigroup generated inverse semigroup monogenic generated single element notation means inverse subsemigroup brandt semigroups section completely inverse semigroups semigroups isomorphic following given group nonempty set consists set products nonzero idempotents therefore correspondence set every primitive inverse semigroup one every nonzero idempotent minimal union brandt semigroups let inverse semigroup let denote principal factor associated formally section least sas practice consider products lie products least defined least kernel group otherwise finite completely thus brandt semigroup general completely semisimple principal factor group brandt semigroup holds contains bicyclic subsemigroup semigroup shall represent bicyclic semigroup via presentation case chain corresponding identity element alternative representations see inverse semigroup whenever let denote least group congruence inverse semigroup maximal group homomorphic image proposition identity relation note inverse semigroup least idempotent say kernel isomorphic main theorem proposition brandt semigroup finitely generated finite finitely generated proof first suppose finite say let generating set usual nonzero nonzero may indexed thus nonzero way group hii isomorphic let transversal identity element green lemma lemma hij clearly finitely generated conversely finitely generated finite since nonzero element either generator inverse generator notation previous paragraph finite subset nonzero products members form gxj gxj nonzero products members hjj hence hbi finitely generated corollary brandt semigroup howson property true proof necessity clear prove converse first consider inverse subsemigroup subgroup primitive thus union brandt subsemigroups clearly finitely generated finitely many factors finitely generated proposition finitely generated factor finitely many idempotents maximal subgroups subgroups finite generated let finitely generated inverse subsemigroups subgroups nonempty first suppose nonzero group subgroup say common subgroup factor factor therefore intersection subgroups preceding paragraph subgroups finitely generated therefore howson property subgroup similarly factor union contained within factor factor intersection two factors maximal subgroups therefore intersection maximal subgroup maximal subgroup therefore finitely generated proposition finitely generated lemma let inverse semigroup hxi inverse subsemigroup hej proof let belongs result hall lemma usual provisions case cited result clearly note xei thus hej let meets nontrivially may regarded inverse subsemigroup denote subgroup put proposition let inverse semigroup finitely many idempotents finitely generated finitely generated meets nontrivially proof generated finite set lemma generated finite set conversely hxj may assume contain zero case interpreted contained hxj thus generated union sets prove theorem stated introduction theorem let inverse semigroup finitely many idempotents howson property maximal subgroups property proof necessity clear converse let finitely generated inverse subsemigroups let meets nontrivially minimum relevant intersections subgroups finitely generated howson property otherwise direct half proposition finitely generated inverse subsemigroups applying proposition finitely generated applying reverse half proposition therefore finitely generated howson property holds finiteness semilattice idempotents necessary remarked introduction free inverse semigroups rank greater one satisfy howson property subgroups trivial corollary let inverse semigroup finitely many idempotents howson property maximal group image property particular theorem semidirect product finite semilattice group howson property group property proof special case minimum isomorphic maximal subgroup embeds result immediate theorem remark corollary silva soares cleverly apply last statement corollary stated extend case locally finite actions orbit element semilattice finitely generated subgroups finite one application actions strongly finite semilattices identity semilattices possessing height function given maximum length chain element identity finitely many elements given height note semilattice idempotents free inverse semigroup finite rank property thus guarantee howson property inverse semigroups general another application corollary case group locally finite moot inverse semigroup whose maximal group homomorphic image locally finite locally finite brown lemma therefore immediately howson property hand remarked introduction corollary may also deduced theorem sufficiency follows carroll theorem inverse semigroup finite semilattice idempotents maximum group homomorphic image embeds semidirect product finite semilattice necessity follows fact isomorphic group kernel monogenic inverse semigroups remarked introduction author trotter showed free inverse semigroup rank one inverse semigroup howson property key tool following result proposition inverse subsemigroup free inverse semigroup rank one inverse subsemigroup generated nonidempotents finitely generated far author aware inverse semigroups hitherto known howson property completely semisimple following results special interest lemma every inverse subsemigroup bicyclic semigroup consist solely idempotents finitely generated proof let subsemigroup let greatest idempotent associated nontrivial nonnegative integer say positive integer every nonnegative integer contains xmr since quotient free inverse semigroup rank one generated finite set nonidempotents together set idempotents suppose contains since xsm direct calculation shows contains xsm xsm case contains every nonnegative previous paragraph thus generated together finite set idempotents corollary bicyclic semigroup howson property proof let finitely generated inverse subsemigroups either consists solely idempotents finite assume otherwise positive integers nonnegative integers replacing respectively necessary may assume contains nonidempotent lemma applies particular case whenever contains idempotent max since nonidempotent otherwise contained finite set max apart free one monogenic inverse semigroups determined following classes defining relations theorem positive integers first class consists finite instances second consists infinite cyclic group ideal extensions group finite rees quotients third consists bicyclic semigroup ideal extensions finite rees quotients proposition every monogenic inverse semigroup howson property proof already noted property first class obvious second class finitely many idempotents maximal subgroups either trivial infinite cyclic theorem applies proved howson property let ideal extension finite rees quotient say let finitely generated inverse subsemigroups proof corollary may assume neither consists idempotents since infinite order neither consists idempotents lemma finitely generated finitely generated since finite also finitely generated references brown interesting combinatorial method theory locally finite semigroups pacific math hall regular semigroups algebra howson intersection finitely generated free groups london math soc jones semimodular inverse semigroups london math soc jones trotter howson property free inverse semigroups simon stevin carroll embedding theorems proper inverse semigroup algebra petrich inverse semigroups wiley new york silva contributions combinatorial semigroup theory thesis university glasgow silva soares howson property semidirect products semilattices groups
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